weighted combinatorial group theory and wild metric complexes w. a. bogley oregon state university...

Weighted Combinatorial Group Theory andWild Metric Complexes

W. A. BogleyOregon State University

A. J. SieradskiUniversity of Oregon

Abstract

In this paper, we develop the low dimensional homotopy theoryrequired for weighted combinatorial group theory. In [S97], the usualconcepts of generators and relators of group presentations are extendedto weighted generators and weighted relators for weighted group presen-tations. This extension parallels the passage from finite sets to ordertypes, i.e. closed nowhere dense sets in the closed unit interval. Inthe weighted environment, products of all order-type are permitted,provided that the entries of the product have weights that limit atzero as their depth of occurrence in the order-type increases withoutbound. Here, we develop weighted analogs of the usual correspondencevia fundamental groups between free groups and 1-dimensional CW cellcomplexes and between group presentations and 2-dimensional CW cellcomplexes. The results are a correspondence between free omega-groupsand wild metric 1-complexes in which the 1-cells can limit on 0-cells anda correspondence between weighted group presentations and wild metric2-complexes in which the 1-cells and 2-cells can limit on the 0-cells.

1991 Mathematics subject classification: Primary 57M20 Secondary57M30, 20F06

1 Path Homotopy in a Metric Bouquet

A basic construction in combinatorial group theory is the free group F (X) withbasis X. Here X denotes a set equipped with an involution −1 : X → X, calledinversion, that fixes exactly one element 1X ∈ X, called the trivial element.The members of the free group F (X) are classes of words in the alphabet X,i.e., finite sequences w = x1 · . . . · xk with xi ∈ X, modulo the equivalencerelation generated by the elementary relations x · x−1 = 1X = x−1 · x andxε ·1X = xε = 1X ·xε for all x ∈ X and ε = ±1. And a basic calculation in low

1

1 PATH HOMOTOPY IN A METRIC BOUQUET 2

dimensional homotopy identifies the free group F (X) with the fundamentalgroup π1(

∨S1x) of the 1-point union (with the weak topology) of 1-spheres S1

x

indexed by the non-identity inverse pairs x = x, x−1 in the basis X.

In this section, we establish a weighted generalization of the correspondencebetween free groups and weak 1-point unions of 1-spheres. In terms introducedin [S97] and re-defined below, we consider the free omega-group Ω(X,wt) witha weighted-basis (X,wt). And we establish an isomorphism of the free omega-group Ω(X,wt) with the fundamental group π1(Z(X,wt)) of a metric 1-pointunion Z(X,wt) of 1-spheres S1

x whose diameters equal the weights wt(x±1)of the corresponding non-identity inverse pairs x = x, x−1 ⊂ X. This isnothing new unless the weighted set (X,wt) has members x ∈ X with weightswt(x) that have limit zero.

1.1 The Motivating Example: The Hawaiian Earring

The weighted vocabulary is motivated by the Hawaiian earring space. Thisis the metric subspace Z of the Euclidean plane E2 consisting of the 1-pointunion of the circles Cn of radii 1

2n, centered on the points ( 1

2n, 0), for n ≥ 1.

For each n ≥ 1, let x±1n : I → Cn ⊂ Z denote the positively- and negatively-

oriented essential circuits of the circle Cn of the Hawaiian earring space Z,beginning and ending at the union point z0 = < 0, 0 >. Let X denote theset of these essential loops x±1

n on the individual circles Cn, together with theconstant loop 1X at the union point z0 = < 0, 0 >. In the terminology of[S97], the metric-weight of a loop f : I → Z, measured from the basepoint z0,in the Euclidean metric subspace (Z,m) is given by:

wt(f) = lubm(z0, f(t)) : 0 ≤ t ≤ 1.

In particular, wt(1X) = 0 and wt(x±1n ) = 1/n for all n ≥ 1. It is well-known

that the fundamental group π1(Z) of the Hawaiian earring space Z is notisomorphic to the free group F (X) with basis X, or indeed to any ordinaryfree group. (See [dS92, G56, M-M86].) There are loop classes that cannot berepresented by a finite product of the essential loops x±1

n : I → Cn ⊂ Z; theseare the loops that take advantage of the limiting behavior of the circles Cnand traverse infinitely many of them in an essential manner. Rather, π1(Z) isisomorphic to the free omega-group Ω(X,wt) with weighted-basis (X,wt), asexplained below.


1.2 Metric Realization of a Weighted Alphabet

We now review the notion of a weighted alphabet (X,wt) and the associ-ated free omega-group Ω(X,wt) [S97] and we construct a metric realizationZ = Z(X,wt) of the weighted alphabet (X,wt) that generalizes the Hawaiianearring.

Let X denote a set equipped with an involution −1 : X → X, called inversion,that fixes an element 1X ∈ X, called the trivial element. A weight functionfor the set with inversion (X, −1) is a real valued function

wt : X →H = 0, . . . , 1n, . . . , 1

2, 1

satisfying the two weight conditions: wt(x) = 0 if and only if x = 1X , andwt(x−1) = wt(x) for all x ∈ X. The pair (X,wt) is called a weighted set. Whenthe inversion −1 : X → X has just the single fixed point 1X , the weighted set(X,wt) is a weighted alphabet.

Metric 1-Point Unions Let X denote the set of non-trivial inverse pairsx = x, x−1 of members of a weighted alphabet (X,wt). For each x ∈ X,let E2

x = E1x × E1

x−1 be a copy of the Euclidean plane E2 and let S1x ⊂ E2

x

denote the circle of radius 12wt(x), passing through the origin, and invariant

under transposition of coordinates. We identify S1x ⊂ E2

x and S1x−1 ⊂ E2

x−1

under the homeomorphism that transposes the Euclidean coordinates and wecall the result the x-circle S1

x ⊂ E2x. Let

E2(X) ⊂∏

x∈XE2x

denote the weak X-product set of all X-tuples a = (ax) that differ from theorigin O = (Ox) for just finitely many indices x ∈ X. Give the weak X-product E2(X) the metric:

m(a, b) = max||ax − bx|| : x ∈ X

using the Euclidean norm || − || in each factor E2x. At the origin O = (Ox),

form the 1-point union ∨S1

x : x ∈ X ⊂ E2(X)

of the x-circles S1x, x ∈ X, in the factor spaces E2

x of the weak X-productE2(X). This metric subspace of the weak product is called the metric realiza-tion of the weighted alphabet (X,wt) and is denoted by Z = Z(X,wt). It isa metric 1-point union of circles that generalizes the Hawaiian earring space.There is an obvious cellular decomposition of Z = Z(X,wt) that involves asingle 0-cell and a single 1-cell for each 1-sphere, but, because the 1-cells limiton the 0-cell, Z does not have the weak topology with respect to this cellulardecomposition. We view Z as a wild metric 1-complex.


1.3 Path Homotopy in the Metric Realization Z(X,wt)

For notational convenience, each non-identity inverse pair x±1 ∈ X is usedto label the positively- and negatively-oriented circuits of the correspondingx-circles S1

x of the metric realization Z = Z(X,wt), beginning and ending atthe union point z0 = (Ox). And the identity element 1X ∈ X is used tolabel the constant loop in Z at z0. Thus we have the loops x±1 : I → Z and1X : I → Z, and in this way treat the set X as a set of labels for specificloops in Z. By construction of the metric realization Z = Z(X,wt), the givenweight wt(x) of each member of (X,wt) equals the metric-weight of its labeledloop x : I → Z, as measured from the basepoint z0 in the Euclidean metricsubspace (Z,m):

wt(x) = lubm(z0, x(t)) : 0 ≤ t ≤ 1.

Words in the Weighted Alphabet (X,wt) An order-type in the closed unitinterval I = [0, 1] is a partition ω : 0 < . . . < 1 by a closed nowhere denseset ω of points that includes the endpoints 0 and 1. The complementarypoint set Uω = [0, 1] − ω is an open subset of the open interval (0, 1) that isdense in the closed interval [0, 1]. So the complement Uω is a countable union⋃i(ai, bi) of disjoint open intervals (ai, bi) in (0, 1) whose closure is [0, 1]. LetIω = i denote the set of the complementary intervals i = (ai, bi) that formthe components of Uω. The complementary intervals i ∈ Iω have lengthsl(i) = bi − ai that sum to 1 and the closure in I of the countable union⋃iai, bi of their end-points is the order-type ω.

Consider a function x : Iω → X, i.e., a choice of elements x(i) ∈ X forall complementary intervals i ∈ Iω, satisfying the metric weight restriction:wt(x(i)) → 0, as l(i) → 0. View the values x(i) as labels that rest on thecomplementary intervals i ∈ Iω of the order-type ω. Taken together, the order-type ω and the labeling function x : Iω → X constitute a (order-type) word(ω, x) for the weighted alphabet (X,wt).

Lemma 1.1 Each order-type word (ω, x) for the weighted alphabet (X,wt) hasan evaluation as a loop p(ω,x) : I → Z in the metric realization Z = Z(X,wt).

Proof: The word (ω, x) assigns to the intervals i ∈ Iω elements x(i) ∈ Xsatisfying the metric weight restriction: wt(x(i)) → 0, as the ω intervals’lengths l(i)→ 0. This assures the continuity of the product path p(ω,x) : I → Zdefined on each ω interval i = (ai, bi) via a re-parametrized version of theassociated loop x(i) : I → Z:

p(ω,x)(t) = x(i)( t−aibi−ai ), for ai ≤ t ≤ bi and all i ∈ Iω, and

p(ω,x)(t) = ∗, otherwise (i.e., for t ∈ ω).


The loop p(ω,x) : I → Z in the metric realization Z = Z(X,wt) is called theword-path associated with the word (ω, x) for the weighted alphabet (X,wt).It spells the word (ω, x). 2

i0 1j

x(i) x( j)

p(ω, x)

ω

Figure 1: Word-path

Normalization of Loops in Z(X,wt) The word-paths p(ω,x) : I → Z forwords (ω, x) for the weighted alphabet (X,wt) represent all path-homotopyclasses [f ] of loops in the metric realization Z = Z(X,wt). To show this,we consider any loop f : I → Z based at the origin z0. The pre-imagef−1(Z−z0) = I − f−1(z0) of the punctured 1-point union Z −z0, beingan open subset of the open interval (0, 1), is a countable union of disjoint openintervals (a, b) ⊂ (0, 1). These intervals have Euclidean lengths b−a with sum0 ≤ σ ≤ 1. When σ = 0, the loop f : I → Z is constant, and so equals theloop p(0<1X<1) = 1X : I → Z associated with the monosyllabic identity word0 < 1X < 1. When 0 < σ ≤ 1, we define the function q : I → I on each pointr ∈ f−1(z0) as 1/σ times the sum of the lengths b−a of the components (a, b)of I − f−1(z0) that lie to the left of the point r. This defines q : I → I on bothendpoints a and b of each component (a, b) of I − f−1(z0), and q is extendedlinearly over each such component. In this way, q : I → I uniformly expandsthe components of I − f−1(z0) by the factor 1/σ to make them disjoint opensubintervals i = (ai, bi) ⊂ I that have lengths l(i) that sum to 1. So q : I → Iis a quotient map that collapses the components of the pre-image f−1(z0) intoindividual points of an order-type that we denote by ω ⊂ I. Let Iω denotethe family of complementary subintervals i ⊂ I for this order-type ω.

Because the given loop f : I → Z is constant on the pre-image f−1(z0), itfactors through the quotient map q : I → I to give a loop f ′ : I → Z. Becausethe maps q, 1I : (I, 0, 1)→ (I, 0, 1) agree on the interval’s endpoints 0 and 1,they are homotopic relative the endpoint subset 0, 1. This fact provides apath-homotopy

f ′ = f ′ 1I ' f ′ q = f : I → Z.

The map f ′ : I → Z carries each complementary subinterval i ∈ Iω for theorder-type ω into some connected component S1

xi− z0 of the punctured


1-point union Z − z0, with its endpoints mapped to the union point z0.So the restriction of f ′ to the closure of the interval i = (ai, bi) ∈ Iω, whenre-parametrized to be a path f ′(i) : [0, 1] → Z, is path-homotopic in S1

x toeither the constant loop 1X : I → Z or to the positively- or negatively-orientedessential circuit x±1

i : I → Z of the circle S1xi

. Whichever is the case, we denotethe representative loop by x(i) ∈ X for each interval i ∈ Iω. Because f ′ iscontinuous, there are at most finitely many appearances of each x ∈ X. In thisway, the original loop f : I → Z determines a labeling function x : Iω → Xfor the order-type ω satisfying the metric weight restriction: wt(x(i))→ 0 asl(i)→ 0. In short, this analysis of the loop f : I → Z yields an order-type ωand word (ω, x) for the weighted alphabet (X,wt).

Lemma 1.2 The loop f : I → Z is path-homotopic to the word-path p(ω,x) :I → Z associated with the word (ω, x) for the weighted alphabet (X,wt).

Proof: As above, we have the path-homotopy f ' f ′ and the path-homotopiesh(i) : I × I → S1

xi⊂ Z for the representative loops f ′(i) ' x(i), i ∈ Iω. The

latter path-homotopies have metric weights, measured from the union pointz0 ∈ Z, that are bounded by the corresponding label weights:

wt(h(i)) = lubm(∗, h(i)(t, s)) : 0 ≤ t, s ≤ 1 ≤ diam(S1xi

) = wt(x(i)).

Because wt(x(i)) → 0 as l(i) → 0, these path-homotopies h(i) can be re-parametrized and stacked side-by-side along their path parameter t to forma path-homotopy H : I × I → Z. In detail, H is defined on each closed ωsubrectangle [ai, bi]× I via a re-parametrized version of h(i) : I × I → Z:

H(t, s) = h(i)( t−aibi−ai , s), for ai ≤ t ≤ bi, 0 ≤ s ≤ 1, and all i ∈ Iω, and

H(t, s) = ∗, otherwise.

This constructs the desired path-homotopy H : f ′ ' p(ω,x) : I → Z. 2

Word Similarity and Path-Homotopy Two words (ω, x) and (ω′, x′) for theweighted alphabet (X,wt) are similar, written (ω, x)

s∼ (ω′, x′), if, when viewedas decoration on the top and bottom edges of a unit square, their intervalsthat are labeled by essential loops x(i) 6= 1X 6= x′(i′) can be simultaneouslypaired (connected) in the unit square by a family of disjoint arcs α, each ofwhich either joins a labeled interval of the word (ω, x) to an identically labeledinterval of the word (ω′, x′) or joins inversely labeled intervals from the sameword (ω, x) or (ω′, x′). We call such a display S of arcs a similarity squarefor the words (ω, x) and (ω′, x′) and we write S : (ω, g) ∼ (ω′, g′). SeeFigure 2(b). An arc whose two ends are at identically labeled intervals inthe two opposing words, say ·x(i)· and ·x′(i′)· where x(i) = x′(i′), is called


an associating arc; an arc whose two ends are at inversely labeled intervalsin the same word, say ·x(j)· and ·x(k)· where x(j) = x(k)−1 or ·x′(j′)· and·x′(k′)· where x′(j′) = x′(k′)−1, is called a cancelling arc. When there are nocancelling arcs, the display is called a associativity square. See Figure 2(a).

The disjoint associating and cancelling arcs just serve to express a compatibil-ity condition for the pairing, and they can even be tightened into disjoint linesegments if one conducts business in a circular disc instead of a square. Asthese arcs join the midpoints of disjoint intervals, they can be made to havedisjoint tubular neighborhoods. And, by compactness considerations, eachtubular neighborhood can be selected to have uniform width in the similaritysquare.

x(i)(ω, x):

(ω′, x′):

(ω, x):

(ω′, x′):x′(i′)

…

…

x(k)

x′(k′)

…

…x′( j′)

x( j)

x′(i′)

x(i)

Figure 2: (a) Associativity Square (b) Similarity Square

We now show that a similarity square S : (ω, x) ∼ (ω′, x′) is equivalent toa path homotopy p(ω,x) ' p(ω′,x′) that has been put in general position withrespect to the centers of the 1-cells of the wild metric 1-complex Z = Z(X,wt).Since so much has already been hidden elsewhere under the guise of generalposition and there is no ready reference for its application in this wild cellularsituation, we present the details:

Theorem 1.3 Two words (ω, x) and (ω′, x′) for the weighted alphabet (X,wt)are similar if and only if the two word-paths p(ω,x), p(ω′,x′) : I → Z are path-homotopic in the metric realization Z = Z(X,wt) of (X,wt).

Proof: The task is to prove that a similarity pairing (ω, g) ∼ (ω′, g′) existsif and only if there is an topological path-homotopy p(ω,x) ' p(ω′,x′).

Well, any given similarity pairing (ω, g) ∼ (ω′, g′) is expressible by a sim-ilarity square S. The disjoint properly embedded associating and cancelling


arcs α : I → S of the similarity square can be made to have disjoint tubu-lar neighborhoods α : I × (−ε,+ε) → S whose ends α(0 × (−ε,+ε)) andα(1 × (−ε,+ε)) are complementary intervals of the two order-types ω andω′ contained in the top and bottom edges of the unit square I × I. The orien-tation conditions on the ends of the associating arcs and cancelling arcs makeit possible to continuously extend the word-paths p(ω,x), p(ω′,x′) : I → Z overthe tubular neighborhoods joining the complementary intervals of ω and ω′.By sending the complement of these tubular neighborhoods to the union pointz0 ∈ Z, there results a path-homotopy H : I × I → Z that establishes thedesired path-homotopy relation p(ω,x) ' p(ω′,x′). This function H is continuousby the weight condition on the words (ω, x) and (ω′, x′) since there are onlyfinitely many of the tubular neighborhoods whose images under H are notcontained in a given neighborhood of the basepoint z0 ∈ Z.

Conversely, given p(ω,x) ' p(ω′,x′), we apply local simplicial approximationtechniques to the path-homotopy H : I × I → Z to derive a similarity squareS : (ω, g) ∼ (ω′, g′), as follows. The complement V = Z − z0 of the unionpoint z0 is an open subset V ⊂ Z which admits a triangulation M in whicheach vertex v has star neighborhood starM(v) with diameter diam(starM (v))in the metric space Z less than or equal to the distance of the vertex v to theboundary point z0 = ∂V . For example, under the obvious identification ofthe punctured x-circle S1

x − z0 with the open interval

(− 12wt(x) ,+

12wt(x) ) ⊂ E1

the vertex set could be ± 2k−12wt(x)2k

: k = 2, 3, . . .. Consider the relativelyopen pre-image U = H−1(V ) in the domain square I × I and the restrictionH : U = H−1(V ) → V of the given homotopy H. Because H is a path-homotopy of word-paths p(ω,x) ' p(ω′,x′) : I → Z, the pre-image U = H−1(V )meets the boundary ∂(I×I) in the complementary intervals i ∈ Iω on I×0and i′ ∈ Iω′ on I × 1. These intervals are assigned essential loop labelsx(i) 6= 1X and x′(i′) 6= 1X in the words (ω, x) and (ω′, x′). Since the word-paths p(ω,x) ' p(ω′,x′) map these open subintervals to circuits of the puncturedcircles in V = Z − z0, the triangulation M of V pulls back under H to atriangulation L of U ∩ (I × I). We extend this over the interior of I × I to atriangulation K of the relatively open subset U = H−1(V ) ⊂ I×I, as follows.

First, we iterate the procedure of subdividing squares into four equal sub-squares, beginning with the unit square I× I. At the k-th stage, there resultsa collection Sk of 22k subsquares of side length 1/2k that tile I × I. As theprocess of subdividing I × I proceeds, we inductively select, for each k ≥ 1,each subsquare S ∈ Sk of side length 1/2k provided: (1) its total enclosure inI × I by all adjacent members of Sk is contained in some member of the opencovering H−1(StarM) of U = H−1(V ) ⊂ I × I by pre-images H−1(starM(v))


of the star neighborhoods starM(v) of vertices v ∈M0 in V = Z − z0, and(2) is not contained in a previously selected square from Sm for m < k. Thisprocedure selects a tiling of the relatively open subset U = H−1(V ) ⊂ I×I bynon-overlapping binary subsquares. By compactness, each selected subsquaremeets just finitely many other selected subsquares. The corner points of thistiling will, in all likelihood, subdivide the 1-simplexes < u, u′ > of the pull-back triangulation L of U ∩ (I × I). But the squares’ corner points that liein I × ∂I have binary fraction coordinates, i.e., rational numbers with form0 ≤ m/2k ≤ 1. By a preliminary isotopy of the word-paths’ domain intervals,we make the endpoints of the complementary open intervals of the order typesω and ω′ and the vertices of their triangulations be binary fractions.

To blend from the tiling to the triangulation L near the boundary of I × I,we extend the homotopy H over I × Iε (Iε = [0 − ε, 1 + ε]) to be stationaryon product neighborhoods I × [−ε, 0] and I × [1, 1 + ε] for the edges inI × I). We extend the tiling of U = H−1(V ) ⊂ I × I using the rectangles[u, u′] × [−ε, 0] and [u, u′] × [1, 1 + ε] for open 1-simplexes < u, u′ > ∈ Lfrom I × 0 and I × 1, respectively. The restriction H : U ∩ (I × I)→ Vis a simplicial map L → K and H is stationary on the product collars. Sothe open-sided rectangle < u, u′ > × [−ε, 0] or [u, u′]× [1, 1 + ε] for an open1-simplexes < u, u′ > of the triangulation L of U ∩ (I × I) belongs to justthe two star neighborhood pre-images H−1(starMv) and H−1(starMv

′) for thevertex images v = H(u) and v′ = H(u′), and the rectangle edge u× [−ε, 0]or u× [1, 1+ε] for a vertex u of L belongs to just the one star neighborhoodpre-image H−1(starMv) for the vertex image v = H(u).

We triangulate the union of the selected subsquares and product rectangles,using their corner points as vertices to give each of their edges a finite sub-division into 1-simplexes, and then subdividing their faces into 2-simplexesvia the join of the barycenters of each face with each of its boundary 1-simplexes. The resulting simplicial complex K is a triangulation of the pre-image of V = Z − z0 under the expanded homotopy H : I × Iε → Z,for which we retain the notation U = H−1(V ), and K has subcomplex Lon U ∩ (I × Iε). By construction, the vertex assignment H : L0 → M0

for the simplicial map H : U ∩ (I × Iε) → V extends to a vertex assign-ment J : K0 → M0 satisfying starK(u) ⊂ H−1(starM(J(u)) for each vertexu ∈ K0. Then the standard simplicial approximation techniques providesa simplicial map J : U = H−1(V ) → V that extends the simplicial mapH : U ∩ (I × Iε) → V and is a simplicial approximation to H : U → V .Because the star neighborhood starM (v) of each vertex for the triangulationM of V = Z − z0 has diameter less than or equal to the distance of thevertex v to the boundary point z0 = ∂(V = Z − z0) in Z, the exten-sion J : I × Iε → Z of J that is constantly z0 off U = H−1(V ), i.e., onH−1(z0) ⊂ I × Iε, is continuous.


So the pre-image J−1(v) of each vertex v ∈ M0 is a closed (hence, compact)subspace of I × Iε, as well as, a subset of the discrete 0-skeleton K0 of U ⊂I × Iε. Therefore, even though the triangulation K of the relatively opensubset U is infinite, each vertex v ∈M0 is the image of at most finitely manyvertices of K; hence, each simplex of M is the image of at most finitely manysimplexes of K. This implies that each component of the pre-image underthis map J : I × Iε → Z of the central 1-simplex in one of the puncturedx-circles S1

x − z0 triangulated by M is a finite simplicial linkage in I × I.(See [S93, Section 1] and Figure 3.) The corresponding component of the pre-image under J of the barycenter of the central 1-simplex is a piecewise linearpath that form the spine α of the simplicial linkage. As in [S93, Theorem 1.4],the collection of those component spines that reach I × Iε qualify I × Iε as asimilarity square S : (ω, x)

s∼ (ω′, x′). Those spines α with end-points in thesame same square edge join intervals of the same word (ω, x) or (ω′, x′) havinginverse labels; those spines α with end-points on opposite square edges joinintervals of the two words (ω, x) and (ω′, x′) having the same label. 2

…

…

(ω′, x′): x′(i′)

(ω, x): x( j) x(k) x(i)

Figure 3: Simplicial linkage and arc spines

1.4 The Word Group and Its Freeness

The Word Group As explained in [S97], similarity is an equivalence relationon the set of words for the given weighted alphabet (X,wt). The constructionof product words via juxtaposition respects the similarity relation for words;indeed, a similarity square for the product arises from the juxtaposition ofsimilarity squares for the factors. In this way, the set of similarity classes


[(ω, x)] of words (ω, x) for the weighted alphabet (X,wt) becomes an associa-tive semigroup Ω(X,wt). The definition of word similarity is rigged to insurethat: (1) any identity word (ω, 1X) is similar to the monosyllabic identity word(0 < 1X < 1), (2) the product (ω, 1X) ·(ω′, x′) of an identity word (ω, 1X) withany word (ω′, x′) is similar to the word (ω′, x′), (3) the product (ω, x) ·(ω, x)−1

of any word (ω, x) and its inverse (ω, x)−1 = (ω, x−1), obtained from the re-versing the order-type and inverting the loop labels, is similar to an identityword. This makes the associative semigroup Ω(X,wt) into a group, called theword group on the weighted alphabet (X,wt).

The preceeding analysis of path-homotopy of loops provides the following com-binatorial description of the fundamental groups of generalized Hawaiian ear-ring spaces.

Theorem 1.4 The fundamental group of the metric realization Z = Z(X,wt)is isomorphic to the word group Ω(X,wt) on the weighted alphabet (X,wt).

J. W. Cannon and G. R. Conner have developed an alternative combinatorialdescription for elements of the fundamental group of the Hawaiian earring andof other one-dimensional spaces [C-C97(1), C-C97(2)]. Cannon and Conneralso consider generalizations of the fundamental group where the unit intervalis replaced by more general linearly ordered spaces [C-C97(3)].

Freeness of the Word Group Just as for an ordinary finite word in anunweighted alphabet, an order-type word (ω, x) for the weighted alphabet(X,wt) has a unique free reduction:

A word (ω, x) for the weighted alphabet (X,wt) is reduced if either it is themonosyllabic identity word 0 < 1X < 1 or there is no subword (ω∗, x∗) of(ω, x) that is similar to an identity word. So a reduced word not similar to anidentity word contains no identity element labels.

Lemma 1.5 ([S97]) Any word (ω, x) for the weighted alphabet (X,wt) is sim-ilar to a reduced word.

A reduced word (ω′, x′) similar to a given word (ω, x) is called a free reductionof (ω, x) in Ω(X,wt). The lemma yield a free reduction (ω′, x′) of each word(ω, x). When (ω, x) is similar to an identity word, it has the unique freereduction 0 < 1X < 1. When (ω, x) is not similar to an identity word, it hasa unique free reduction in this sense:

Lemma 1.6 ([S97]) Any two free reductions (ω′, x′) and (ω′′, x′′) of a word(ω, x) for the weighted alphabet (X,wt) are associated in this sense: there isa similarity square S : (ω′, x′)

a∼ (ω′′, x′′) free of conjugation arcs.


It is shown in [S97] that the word group Ω(X,wt) has its own weight function,the word-weight function

w-wt : Ω(X,wt)→ [0,∞)

that assigns to each similarity class [(ω, x)] the maximal weight wt(x(i))achieved by a label x(i) of the reduced representative (ω, x) of the class. Asa weighted group, the word group (Ω(X,wt), w-wt) is an omega-group (see[S97]) and satisfies the universal freeness property in the restricted categoryof omega-groups (G,wt):

Theorem 1.7 ([S97]) Consider the word group Ω(X,wt) on any weighted al-phabet (X,wt).

1. The word-weighted word group (Ω(X,wt), w-wt) is an omega-group: anyword (ω, [(λ, x)]) for the weighted word group (Ω(X,wt), w-wt) can beevaluated as the similarity class of the composite word (ω λ, x).

2. Universal Property: A weighted function φ : (X,wt) → (G,wt) to anomega-group (G,wt) (i.e., φ(1X) = 1G, φ(x) = φ(x−1) for all x ∈ X, andwt(φ(x)) → 0 whenever wt(x) → 0) determines a unique omega-grouphomomorphism φ : (Ω(X,wt), w-wt)→ (G,wt) with the assigned valuesφ([(τ, x)]) = φ(x) on all the monosyllabic words (τ, x) = (0 < x < 1).

For this reason, the omega-group (Ω(X,wt), w-wt) is called the free omega-group with weighted-basis (X,wt). By Lemmas 1.5 and 1.6, we may view themembers of the free omega-group as reduced words (ω, x) in the weightedbasis (X,wt). By the construction in part (1) of Theorem 1.7, order-typeproducts in the free omega-group can be expressed in terms of reduced wordsvia composition followed by free reduction. In detail: the order-type productω〈(λ, x)〉 ∈ Ω(X,wt) is defined whenever the reduced words (λ(i), x(i)) ∈Ω(X,wt), i ∈ Iω, have word-weights

w-wt((λ(i), x(i))) = maxwt(x(i)(j)) : j ∈ Iλ(i)

that tend to zero as the ω intervals’ lengths l(i) tend to zero. This order-typeproduct ω〈(λ, x)〉 is the free reduction of the composite word (ω λ, x) havingsubwords (λ(i), x(i)) on the intervals i ∈ Iω of the order-type ω.

2 REALIZATIONS OF WEIGHTED PRESENTATIONS 13

2 Realizations of Weighted Presentations

We now define weighted group presentations and their metric realizations.

2.1 Weighted Presentations and their Groups

The weighted versions of groups presentations and the groups they presentwere introduced in [S97]. Here is a review of this terminology.

Weighted Presentations A weighted group presentation

P = < (X,wt) : (R, r-wt) >

consists of a weighted generator set (X,wt) and a weighted relator set (R, r-wt)(both with weights in H = 0, . . . , 1

n, . . . , 1

2, 1). The former gives

rise to the free omega-group (Ω(X,wt), w-wt) with weighted-basis (X,wt).The latter is comprised of reduced words r = (γr, xr) and their inverses r−1

from (Ω(X,wt), w-wt) that are assigned relator-weights r-wt(r) greater than orequal their word-weights w-wt(r) derived from their constituent generators inX: r-wt(r) ≥ w-wt(r). This inequality is called the relator-weight condition.For notational convenience, the identity element 1X ∈ X is included in Rwhere it is denoted by 1R and is assigned trivial relator-weight r-wt(1R) = 0.

The relator-weight condition is designed to permit application of the weightedpresentation vocabulary to the fundamental group of metric cell complexeswhose 2-cells may have diameters that exceed that of their boundary.

Consider any order-type λ and a pair of words (λ,w) and (λ, r) for the twoweighted sets (Ω(X,wt), w-wt) and (R, r-wt), respectively. These λ-wordsinvolve labels w(i) ∈ Ω(X,wt) and r(i) ∈ R, for the intervals i ∈ Iλ, hav-ing word-weights w-wt(w(i)) and relator-weights w-wt(r(i)) ≤ r-wt(r(i)) thattend to zero as the λ intervals’ lengths l(i) → 0. The word-weights of theproduct labels w(i) · r(i) ∈ Ω(X,wt), i ∈ Iλ, satisfy the restriction:

w-wt(w(i) · r(i)) ≤ maxw-wt(w(i)), w-wt(r(i))

and so also tend to zero as l(i) → 0. So (λ,w · r) is a second word for theweighted set (Ω(X,wt), w-wt). Because (Ω(X,wt), w-wt) is an omega-group,the word evaluations λ〈w · r〉 and λ〈w〉 exist in (Ω(X,wt), w-wt).

Weighted Consequences A weighted consequence in the free omega-group(Ω(X,wt), w-wt) of the weighted relator set (R, r-wt) is any difference product

λ〈w · r〉 · (λ〈w〉)−1 ∈ Ω(X,wt),


associated with an order-type λ and a pair of words (λ,w) and (λ, r) for thetwo weighted sets (Ω(X,wt), w-wt) and (R, r-wt), respectively.

Weighted Normal Closure The weighted normal closure of the weightedrelator set (R, r-wt) in the free omega-group (Ω(X,wt), w-wt) is the groupN(R, r-wt) of all weighted consequences

λ〈w · r〉 · (λ〈w〉)−1 ∈ Ω(X,wt)

of (R, r-wt) in (Ω(X,wt), w-wt).

Presented Groups The weighted group presentation P presents the group

Π(P) = Ω(X,wt)/N(R, r-wt),

the quotient of the free omega-group (Ω(X,wt), w-wt) on the weighted alpha-bet (X,wt) modulo the weighted normal closure N(R, r-wt) of the weightedrelator set (R, r-wt). (Warning: The presented group need not be an omega-group; see [S97] and [B-S97(2)].)

2.2 Metric Realization of a Weighted Presentation

Let P = < (X,wt) : (R, r-wt) > be any weighted presentation. The metricrealization Z(X,wt) of the weighted generator set (X,wt) is the 1-point union∨

S1x : x ∈ X ⊂

∏x∈XE

2x

of a collection of x-circles S1x (x ∈ X) each constructed to have its diameter

equal to the weight wt(x) = wt(x−1). Similarly, the weighted relator set(R, r-wt) gives rise to a collection of 2-balls having their diameter regulatedby the relator-weights. Let R be the set of non-trivial inverse pairs r = r, r−1in (R, r-wt). For each r ∈ R, let E3

r = E1r ×E1

r−1 ×E1 be a copy of Euclidean3-space E3 and let D2

r ⊂ E3r denote the horizontal r-disc with:

radius : 1√2r-wt(r), height : 1√

2r-wt(r), centerpoint : (0, 0, 1√

2r-wt(r)).

We identify D2r ⊂ E3

r and D2r−1 ⊂ E3

r−1 under the homeomorphism that trans-poses the first two Euclidean coordinates and we call the result D2

r ⊂ E3r . Let

P = X ∪ R and let

E(P) ⊂ (∏

x∈XE2x)× (

∏r∈RE

3r )

denote the weak P-product set comprised of all P-tuples a = (aj) that differfrom the origin O = (Oj) for just finitely many indices j ∈ P = X ∪ R. Givethe weak P-product E(P) the metric:

m(a, b) = max||aj − bj|| : j ∈ P.


View the factor spaces E2x and E3

r as subspaces of the weak P-product E(P)comprised of P-tuples a = (aj) that differ from the origin O = (Oj) in justcoordinates ax ∈ E2

x and ar ∈ E2r , respectively. So E(P) contains the metric

realization Z(X,wt) =∨S1x of the weighted generator set (X,wt), as well as

the horizontal r-discs D2r , r ∈ R. Each of these discs has been sized to meet

the metric sphere

S(O, r-wt(r)) = a ∈ E(P) : m(O, a) = r-wt(r),

centered on the origin O = (Oj), in its boundary circle S1r = ∂D2

r .

Each relator r ∈ R is a reduced word r = (γr, xr) for the weighted generatorset (X,wt). By Section 1, the relator word (γr, xr) spells a word-path p(γr ,xr) :I → Z(X,wt) into the metric realization Z(X,wt) =

∨S1x of (X,wt). Being

a loop, this word-path induces a boundary map

φr : ∂D2r →

∨S1x.

To construct a metric realization of the weighted presentation P, it remainsto use these boundary maps to connect the boundary circle ∂D2

r of each r-discD2r (r ∈ R) to Z(X,wt) =

∨S1x.

Metric Realization of P The metric realization Z(P) of the weightedpresentation P is the metric subspace of E(P) comprised of the x-circles S1

x ⊂E2x (x ∈ X) the horizontal r-discs D2

r ⊂ E3r (r ∈ R, together with the line

segment [a, φr(a)] in E(P) joining each boundary point a ∈ ∂D2r with its

image point φr(a) ∈ ∨S1x. The origin O = (Oj) of the weak P-product E(P)

is taken as the basepoint z0 of Z(P).

The relator word r = (γr, xr) assigns to any interval i ∈ Iγr of the order-typeγr some label xr(i) ∈ X. The boundary map φr : ∂D2

r →∨S1x maps the

corresponding arc in S1r to an oriented circuit of the 1-sphere S1

xr(i)in∨S1x.

So the line segments [a, φr(a)] for points a ∈ ∂D2r of the corresponding arc

consist of linear combinations

t · a+ (1− t) · φr(a) ∈ E2xr(i) ×E

3r , 0 ≤ t ≤ 1.

Keep in mind that a ∈ ∂D2r is really a P-tuple a = (aj) that differs from the

origin O = (Oj) in just the coordinate ar ∈ ∂D2r ⊂ E2

r and that φr(a) ∈ E2xr(i)

is really a P-tuple φr(a) = (φr(a)j) that differs from the origin O = (Oj) injust the coordinate φr(a)xr(i) ∈ S1

xr(i) ⊂ E2xr(i)

.

Two line segments [a, φr(a)] and [a′, φr′(a′)] in Z(P) intersect at the points

t · a+ (1− t) · φr(a) ∈ E2xr(i) ×E

3r , 0 ≤ t ≤ 1.

andt′ · a′ + (1− t′) · φr′(a′) ∈ E2

x′r′(i′) × E3

r′, 0 ≤ t′ ≤ 1


if and only if either (1) r 6= r′, in which case t = 0 = t′ and the intersectionpoint belongs to S1

xr(i)∩S1

x′r′(i′) which is either S1

xr(i)= S1

x′r′(i′) or the union point

O = (Oj), or (2) r = r′, in which case t = t′ (by comparison of the points’height coordinate in E3

r = E3r′) and a = a′ (by comparison of the points’ other

two coordinates in E3r = E3

r′), so that the line segments are the same. Thisproves that the union of each horizontal r-disc D2

r with the open line segments[a, φr(a)) for boundary points a ∈ ∂D2

r , is the interior of an adjoined 2-ballB2r whose boundary S1

r = ∂B2r has been glued to

∨S1x under the boundary

map φr : S1r ≡ ∂D2

r →∨S1x. Moreover, it proves that the interiors of these

adjoined 2-balls B2r , r ∈ R, are disjoint in Z(P) ⊂ E(P).

The basepoint z0 serves as a 0-cell c0, the punctured 1-spheres S1x−z0, x ∈ X,

are topological 1-cells c1x and the interiors Int(B2

r ) of the adjoined 2-ballsB2r , r ∈ R are topological 2-cells c2

r. But the metric realization Z(P) does nothave the weak topology with respect to the cellular decomposition

Z(P) = c0 ∪ c1x : x ∈ X ∪ c2

r : r ∈ R.

Rather, e.g., in the case that there are just finitely many generators and re-lators of each weight, a subset F ⊂ Z(P) is closed in Z(P) if and only ifF ∩ c is closed in each closed cell c and F contains c0 whenever infinitely manyintersections F ∩ c with the open cells c are non-empty. Since there are fewerclosed subsets than the cellular topology would admit, there are more pathsand path-homotopies in Z(P) than cellular homotopy theory would anticipate.On the other hand, there are fewer continuous functions on Z(P) and Z(P)×Ithan the cellular topology would admit; continuity on the individual closedcells alone does not guarantee global continuity. This fact complicates theinvestigation of homotopy in this metric 2-complex Z(P), but it makes it pos-sible for the metric realization Z(P) to have its fundamental group π1(Z(P))presented by the weighted presentation P.

Here is the crucial property of the metric 2-complex Z(P) that enables oursubsequent investigations:

Lemma 2.1 The identity map of the metric realization Z(P) is homotopicrelative the one-point union

∨S1x to a map that retracts into

∨S1x the comple-

ment in Z(P) of the union of the horizontal r-disc interiors Int(D2r), r ∈ R.

Proof: The homotopy H : Z(P)× I → Z(P) (rel∨S1x) moves each point

t · a+ (1− t) · φr(a) ∈ B2r − Int(D2

r)

along its line segment [a, φr(a)] to its endpoint φr(a) ∈ ∨S1x:

H(t · a+ (1− t) · φr(a), s) = (1− s) · (t · a+ (1− t) · φr(a)) + s · φr(a)),


and H radially expands each horizontal r-disc D2r onto the entire adjoined

2-ball B2r . Each restriction H : c2

r × I → Z(P) is continuous and each open2-cell c2

r is open in Z(P). So H is seen to be continuous at each point of c2r×I.

By the relator weight condition, r-wt(r) ≥ w-wt(r), each 1-cell c1x (x ∈ X)

bounds just 2-cells c2r (r ∈ R) whose central discs have height 1√

2r-wt(r) ≥

1√2wt(x). So points on lines segments in Z of height less than ε < 1√

2wt(x)

emanating from relative neighborhoods in c1x define a local basis Nε(z) for

each point z of c1x with the property that the deforming homotopy H retracts

each neighborhood Nε(z) over itself to Nε(z) ∩ c1x. This assures continuity

of H at each point of c1x × I. Again by the relator weight condition, each

closed 2-cell c2r is contained in the closed metric ball of radius r-wt(r) in Z(P)

centered on the origin O = (Oj), which is the sole 0-cell c0. So points onlines segments in Z of heights less than 1√

2ε emanating from ε-neighborhoods

in∨S1x of c0 together with entire 2-cells c2

r for r-wt(r) < ε define a localbasis Nε(z0) for z0 = c0 with the property that the deforming homotopy Hdeforms each neighborhood Nε(z0) over itself off each 2-cell that it doesn’tentirely contain. This assures the continuity of H at each point of c0 × I. 2

The metric complex Z(P) has closed skeleta, the k-cells are open in the k-skeleton for k = 1, 2, and the usual technique of radially expanding centralportions of the 2-cells defines a deforming homotopy H : Z(P) × I → Z(P)of the identity map 1Z(P) relative the 1-skeleton. Moreover this deforminghomotopy H : Z(P)×I → Z(P) respects neighborhood bases in the sense thateach point z in the 1-skeleton has a local basis of neighborhoods Nε(z) suchthat H(Nε(z) × I) ⊆ Nε(z). This is a standard feature for CW 2-complexes,but is available here only because the 1- and 2-cells of Z(P) limit just on the0-cell. We call such a space a wild metric 2-complex.

2.3 Paths and Path-Homotopies in Z(P)

First of all, the word-paths p(ω,x) : I → Z(X,wt) ⊂ Z(P) for words (ω, x)for the weighted alphabet (X,wt) represent all path-homotopy classes [f ] ofloops in the metric realization Z(P).

Lemma 2.2 Any loop f : I → Z(P) at z0 is path-homotopic to the word-path p(ω,x) : I → Z(P) associated with a word (ω, x) for the weighted alphabet(X,wt).

Proof: The pre-image under the loop f : I → Z(P) of the union⋃r∈R Int(D2

r)of the interiors of the horizontal r-discs, being an open subset of the openinterval (0, 1), is a countable union of disjoint open intervals (a, b) ⊂ (0, 1).


The loop f : I → Z carries each such subinterval (a, b) into some connectedcomponent Int(D2

r) of the union, with its endpoints mapped to the boundarycircle S1

r . So the restriction of f to the closed interval [a, b] is a path segmentthat is path-homotopic in D2

r to a path in the boundary S1r . Because the

diameter of each such straight-line path-homotopy is bounded by the distancem(f(a), f(b)) in Z(P) between the end-points of the path segment, these path-homotopies can be stacked side-by-side along their path parameter to form apath-homotopy f ' f ′ to a path f ′ in Z(P) that misses

⋃r∈R Int(D2

r). Thenby Lemma 2.1, there is a path-homotopy f ′ ' f ′′ to a path in the 1-point union∨S1x. Then by Lemma 1.2, it further deforms to a word-path p(ω,x) : I → Z(P)

associated with a word (ω, x) for the weighted alphabet (X,wt). 2

…

(ω′, x′)

(ω, x)…

Figure 4: Similarity Square Modulo (R, r-wt)

Similarity Modulo (R, r-wt) Let two reduced words (ω, x) and (ω′, x′)decorate the top and bottom edges of a unit square. An oriented disc in thatunit square, with its counter-clockwise boundary circuit decorated by a cyclicversion of a relator word r = (λr, xr) ∈ R, is called a relator disc. The tworeduced words (ω, x) and (ω′, x′) for the weighted generator set (X,wt) aresimilar modulo (R, r-wt), written

(ω, x)s∼ (ω′, x′) (mod R),

if there is a countable family of disjoint relator discs in the interior of theunit square such that: (1) the relator disc diameters tend to zero if and onlyif their relator-weights tend to zero, and (2) the intervals of the opposingedges of the square and the boundaries of the relator dics that are labeled bynon-identity elements of (X,wt) can be simultaneously paired (connected) in


the complement of the relator discs in the unit square by a family of disjointarcs, preserving labeling and orientation. That is, paired intervals in the sameboundary component or in boundary of different relator discs are inverselylabeled; all other paired intervals are identically labeled. The arcs and discsare not permitted to accumulate on one another (although they may accumu-late in the square); equivalently, the discs and the arcs have disjoint tubularneighborhoods in the unit square. And, by compactness considerations, eachtubular neighborhood can be selected to have uniform width in the similaritysquare. We call such a display S of arcs a similarity square modulo (R, r-wt)for the reduced words (ω, x) and (ω′, x′). See Figure 4. By the relator-weightrestriction on each relator and the weight restriction on words, there are justfinitely many relator discs and arcs in any similarity square S whose labelshave weights bounded away from zero by a given bound ε > 0.

The similarity relation, modulo (R, r-wt), respects the product of two wordsand so the quotient set of equivalence classes is a group.

Our aim is to show that a similarity square S : (ω, x) ∼ (ω′, x′) (modR)is equivalent to a path homotopy p(ω,x) ' p(ω′,x′) in Z(P) that has been putin general position with respect to the centers of the 1-cells and 2-cells of thewild metric complex Z = Z(P). The first step is to establish the followingweighted version of van Kampen’s theory of group diagrams:

Theorem 2.3 The group Π(P) presented by a weighted group presentation

P = < (X,wt) : (R, r-wt) >

is the group of similarity classes, modulo the weighted relator set (R, r-wt), ofreduced words (ω, x) for the weighted generator set (X,wt).

Proof: The claim is that there is a similarity square (ω, x)s∼ (ω′, x′) (mod R)

if and only if [(ω, x)]·N = [(ω′, x′)]·N , where N is the weighted normal closureN(R, r-wt). Since the similarity relation respects the multiplication of words,it is equivalent to show (ω, x)

s∼ 1X (mod R), if and only if [(ω, x)] ∈ N . Forthis purpose, let S be a similarity disc modulo (R, r-wt) with boundary word(ω, x). We seek a weighted consequence (λ,w · r) · (λ,w)−1 of (R, r-wt) thatis (freely) similar to (ω, x).

The similarity disc S has finite subpictures Sn consisting of arcs and discswith weights not less than 1/n. We construct a sequence of finite trees

T0 ⊂ T1 ⊂ . . . ⊂ Tn ⊂ Tn+1 ⊂ . . .

with the following properties:


(a) T0 is just the basepoint vertex b0 of the ambient disc.

(b) the complement Tn − Tn−1 has two types of vertices:

(i) interior vertices: one vertex vρ in each region ρ of Sn ∪ Tn−1, and

(ii) basepoint vertices: the basepoints br of the relator discs dr of thecomplement Sn − Sn−1.

(c) The edges e of the complement Tn − Tn−1 arise from

(i) a maximal forest in the dual graph to Sn relative to Tn−1 ∪ Sn−1,

(ii) connecting edges: one edge for each component of the above maxi-mal forest connecting it within a region of Sn ∪ Tn−1 to an interiorvertex of Sn−1, and

(iii) terminal edges: one edge er for each relator disc dr connecting itsbasepoint br within a region of Sn ∪ Tn−1 to an interior vertex ofthe complement Tn − Tn−1.

(d) The vertex set of the union T =⋃Tn is discrete in the metric topology

(NB. This uses the assumption that the relator discs have disjoint regularneighborhoods.) The interior vertices can be selected to be outside theclosure of the arcs and discs of S.

(e) Each edge e of T crosses the arcs of S tranversely and determines order-type crossing word we obtained by reading arc crossings as the edge istraversed in the direction toward the basepoint b0 of the ambient disc.

When one views the planar tree T of labeled edges e and labeled boundarycircuits of the R-discs from the outside a tubular neighborhood of their union,one sees a parallel copy of each edge e on each side of its tubular neighbor-hood, and one sees an expanded copy of the boundary of each relator-disc d2

r.Both copies of the edge e support a crossing word we obtained by reading arccrossings encountered as the edge is traversed in the direction away from thebasepoint, and the copy of the boundary of a r-disc supports the relator wordr. Taken together, in their order around the planar tree T , these words forman order-type word (λ,w · r) for which (λ,w) is similar to an identity word.We argue this as follows.

The complementary intervals of the order-type λ ⊂ I consist of a pair ofintervals i−e and i+e for each edge e of T , and an interval ir for each relatordisc dr. The order of these intervals in I is regulated by these rules: for eachedge e, i−e < i+e ; if e is a terminal edge that ends at the relator disc dr, theni−e < ir < i+e and these intervals are consecutive in I; if the removal of theedge e separates e′ from the basepoint b0, then i−e < i−e′ < i+e′ < i+e ; otherwise,


the removal of some vertex separates e and e′ from one another and fromthe basepoint b0, and then i−e < i+e < i−e′ < i+e′ if the terminal branch of thetree containing e lies to the right of that containing e′ when viewed fromthat vertex. The interval i−e is labeled with the word w−1

e read from the arcsencountered along a traverse of the edge e away from the basepoint b0; theinterval i+e is labeled with the reverse word we read from the arcs encounteredalong a traverse of the edge e toward the basepoint b0; and, the interval ir islabeled with its relator word r.

These word labels have the same shape w(i) · r(i) for all intervals i ∈ Iλ, ifone interpretes the three cases i = i−e , i

+e , ir to read w(i) · r(i) = w−1

e · 1R, we ·1R, 1X ·r, respectively. Thus the resulting λ-word can be denoted by (λ,w ·r),and can be interpreted as the boundary word of the union of the tree and therelator discs. The tree T cuts the arcs of the given similarity disc picture Sinto disjoint pieces that provide a pairing of the entries of the boundary word(λ,w ·r) with those of the original disc boundary word (ω, x). The disjointnessof these planar arcs and the construction of the order-type λ imply that anytwo pairs are either nested or disjoint in the difference (λ,w · r) · (ω, x)−1. Inother words, there is a similarity relation (ω, x)

s∼ (λ,w · r).Because any two interval pairs, i−e , i+e and i−e′, i+e′, for the order-type λ areeither nested (ie− < ie′− < ie′+ < ie+) or are disjoint (ie− < ie+ < ie′− < ie′+),and because these pairs are labeled with inverse words, w(e)−1, w(e) andw(e′)−1, w(e′), the λ-word (λ,w) is similar to the identity word (λ, 1X).Thus the similarity relation above shows (ω, x)

s∼ (λ,w · r) · (λ,w)−1.

This proves that when (ω, x) ∼ 1X mod R, then (ω, x) represents the similarityclass of a weighted consequence (λ,w · r) · (λ,w)−1, i.e., [(ω, x)] ∈ N(R, r-wt).The converse holds, since for each weighted consequence (λ,w · r) · (λ,w)−1

there is an obvious similarity square S : (λ,w · r) s∼ (λ,w)−1 mod R. 2

A cautionary note: The tree decorated with relator discs constructed aboveneedn’t be achieved by a path in the similarity disc if the relator discs andarcs are badly displayed, yet this tree does determine the continuous word-path p(λ,w·r) : I → Z into the model Z = Z(P).

The preceding weighted version of the van Kampen theory of group diagramsis one half of the calculation of fundamental groups of the metric realizationsof weighted presentations. Here is the second half:

Theorem 2.4 Two words (ω, x) and (ω′, x′) for the weighted alphabet (X,wt)are similar modulo the weighted relator set (R, r-wt) if and only if the word-paths p(ω,x), p(ω′,x′) : I → Z(X,wt) are path-homotopic in the metric realizationZ = Z(P) of the weighted presentation

P = < (X,wt) : (R, r-wt) > .


Proof: A similarity square S : (ω, g)s∼ (ω′, x′) (mod R), converts into a

topological path homotopy p(ω,x) ' p(ω′,x′) in Z(P) just as for a similar square(see Theorem 1.3). For the converse, letH : I×I → Z(P) be a path-homotopyp(ω,x) ' p(ω′,x′) of word-paths. As in the proof of Theorem 1.3, we apply localsimplicial approximation techniques to derive a similarity square modulo theweighted relator set (R, r-wt)

S : (ω, g)s∼ (ω′, x′) (mod R),

as follows. The complement in Z(P) of the 1-point union∨S1x is the union

W =⋃r∈R Int(B2

r ) of the interiors of the attached 2-balls B2r . Because the

attached 2-balls accumulate in Z(P) only at the deleted union point, the unionW has the weak topology with respect to the interiors of the attached 2-balls.This open set W admits a triangulation N for which the star neighborhood,starN (v), of each vertex v ∈ N0 has diameter in Z(P) less than or equalto the distance of the vertex v to the boundary ∂W ⊂ ∨

S1x and for which

each horizontal r-disc D2r lies in the interior of a 2-simplex s2

r ∈ N and iscentered on its barycenter b(s2

r). The given homotopy H has a restrictionH : Q = H−1(W )→ W to the relatively open subset

Q = H−1(W ) = (I × I)−H−1(∨S1x)

of the domain square. Since H is a path-homotopy p(ω,x) ' p(ω′,x′) : I → Z ofword-paths in

∨S1x, the pre-image Q = H−1(W ) misses the boundary ∂(I×I).

Then (using techniques of proof of Theorem 1.3) we construct a triangulationK of the relatively open subset Q ⊂ Int(I×I), whose stellar covering, StarK,refines the open covering H−1(StarN) of Q by pre-images H−1(starN (v)) ofstar neighborhoods starN (v) of the vertices v ∈ N0 in W .

Standard simplicial approximation techniques provide a simplicial map J :Q = H−1(W ) → W . Because each star starN (v) for the triangulation N ofW has diameter less than or equal to the distance of the vertex vertex v ∈ N0

to the boundary ∂W ⊂ ∨S1x, the extension J : I × I → Z that agrees with

H : I × I → Z off Q = H−1(W ), i.e., on H−1(∨S1x), is continuous. So the

pre-image J−1(v) of each vertex v ∈ N0 is a closed subset of I × I, as wellas, a subset of the discrete 0-skeleton K0 of Q ⊂ Int(I × I). Therefore, eventhough the triangulation K of the relatively open subset Q is infinite, eachvertex v ∈ N0 is the image of at most finitely many vertices of K; hence,each simplex of N is the image of at most finitely many simplexes of K. Thisimplies that each component of the pre-image under this map J : I × I → Zof the horizontal r-disc D2

r in the interior of the central 2-simplex s2r ∈ N is a

finite union of topological discs in Int(I × I). By the radial expansion of eachhorizontal r-disc D2

r onto the attached disc B2r provided by Lemma 2.1, the

new homotopy J : I×I → Z(P) deforms relative ∂(I×I) to one for which thepre-images of the interior Int(B2

r ) of each attached 2-ball B2r in Z(P) is a finite


union of open 2-discs d2r,j whose closures are disjoint discs in Int(I × I) that

are mapped characteristically to B2r . The radial expansion process also makes

the restriction of J : I × I → Z(P), on a suitably oriented boundary circuitof each open 2-disc d2

r,j, read as the word-path p(γr ,xr) : I → ∨S1x spelling the

relator r = (γr, xr).

The union of the open discs d2r,j ⊂ Int(I × I), r ∈ R, can accumulate only at

points of J−1(z0), because the images of a convergent sequence of their centralpoints that is not eventually constant is a convergent sequence of central pointsof the attached 2-balls B2

r . So these open discs qualify as the system ofrelator discs in a similarity square modulo (R, r-wt). For convenience in thesecond stage of the argument, we shape the open 2-discs d2

r,j into open binary

subsquaresI2r,j of Int(I × I). Then the path-homotopy J : I × I → Z(P)

carries the complement of these open binary subsquares into the one-pointunion

∨S1x. Then we repeat the homotopy analysis in the proof of Theorem

1.3, this time for the restriction J : (I × I) − (⋃ I2r,j) →

∨S1x and this time

relative the pre-image U = J−1(V ) of the complement V = (∨S1x)−z0. We

use the same triangulation M of V , but this time pull it back under J to atriangulation L of the intersection of U with the collective boundary

∂(I × I) ∪ (⋃∂I2

r,j).

The rest of the argument proceeds as in the proof of Theorem 1.3 and the

result is a system of arcs in the complement (I × I)− (⋃ I2r,j) that completes

the system of relator squares I2r,j into a similarity square modulo (R, r -wt):

S : (ω, g)s∼ (ω′, x′) (mod R).

2

Theorems 2.3 and 2.4 combine to yield:

Theorem 2.5 The weighted group presentation

P = < (X,wt) : (R, r-wt) >

presents the fundamental group π1(Z) of its metric realization Z = Z(P).

3 EXAMPLES 24

3 Examples

We give several examples of metric realizations of weighted presentations.Similarity square analyses of the presented groups reveal algebraic phenomenathat are peculiar to the world of weighted combinatorial group theory.

Example 3.1 A disc of crescents between rings of the Hawaiian earring.

The weighted presentation PDC :

< (xn, wt 1n) : n ≥ 1 : (rn := xn · x−1

n+1, r-wt1n) : n ≥ 1 >

presents the trivial group. The metric realization of PDC is an assembly oflunar crescents between the consecutive rings of a copy of the Hawaiian earringHE, metrized to be homeomorphic to a disc D bounded by the largest ring.The triviality of the presented group is easily visualized since any loop deformsover the individual crescents bounded by the consecutive rings to their unionpoint; equivalently, any word (ω, x) is similar modulo (R, r-wt) to the identityword (ω, 1X) via the similarity square (see Figure 5) that stacks below anyappearance of a generator xn in (ω, x) an entire sequence of relator discs ofvanishing size for the corresponding sequence of relators rn, rn+1, . . . .

xn

…

rn

rn+1

rn+2

rn+3

rn+4

1X

Figure 5: Similarity in Disc of Crescents

Example 3.2 A harmonic onion whose expanding hemispherical layers havecircular boundaries of vanishing diameter.

The weighted presentation PHO:

< (xn, wt 1n) : n ≥ 1 : (rn := xn, r-wt 1) : n ≥ 1 >

3 EXAMPLES 25

presents the quotient group Π(PHO) of the free omega-group Ω(X,wt) on theweighted basis

(X,wt) = (1X , wt 0) ∪ (xn, wt 1n) : n ≥ 1,

modulo the weighted normal closure N(R, r-wt) of the weighted relator set

(R, r-wt) = (1R, r-wt 0) ∪ (rn := xn, r-wt 1) : n ≥ 1.

Because all the relator weights are 1 and the relators rn are the individualgenerators xn, the weighted normal closure is the usual normal closure N =NΩ(X,wt)(F (X)) in the free omega-group Ω(X,wt) of the ordinary free groupF (X) on the set X of generators. The cosets (ω, x) ·N in this quotient groupΠ(PHO) are called germs of words (ω, x) for the weighted alphabet (X,wt)and two reduced words (ω, x) and (ω′, x′) determine the same germ if andonly if they become similar when finitely many of the generators x ∈ X arereplaced by the trivial element 1X . The feature that (X,wt) is a weightedbasis for Ω(X,wt) is lost in the formation of the quotient group; the cosetsx ·N, x ∈ X, are all the trivial germ 1 ·N and so do not generate the quotientgroup in any sense. Although all the members of the weighted generator setbecome trivial in the presented group Π(PHO), this group is not trivial. Thisconfusing state of affairs is part of the failure of Π(PHO) to be an omega-group:order-type products are not well-defined in this quotient group ([S97]).

…

…

Figure 6: Harmonic Onion

The metric realization HO of PHO is a 1-point union of a sequence hemispher-ical discs whose diameters are bounded away from zero, but whose boundarieshave diameters that limit on zero. This space HO can be embedded as anonion in Euclidean 3-space. First form the 1-point union of 2-spheres Sn ofincreasing diameter 2− 1

ncentered on (1− 1

2n, 0, 0). The generators xn corre-

spond to the circles Cn of intersection of the expanding spheres Sn with the

3 EXAMPLES 26

planes z = nx that contain the y−axis and have slope n. The relators rn := xncorrespond to the larger hemispherical caps Hn on the 2-spheres Sn that liebelow the circles Cn. The generator circles Cn shrink and limit on the unionpoint (0, 0, 0), while the relator hemispheres Hn expand. The 1-point unionHO = ∨nHn is called the harmonic onion. See Figure 6.

Example 3.3 The harmonic archipelago bounded by the Hawaiian earring.

The weighted presentation PHA:

< (xn, wt 1n) : n ≥ 1 : (rn := xn · x−1

n+1, r-wt 1) : n ≥ 1 >

presents the quotient group Π(PHA) of the free omega-group Ω(X,wt) on thesame weighted basis (X,wt) modulo the weighted normal closure N(R, r-wt)of the weighted relator set

(R, r-wt) = (1R, r-wt 0) ∪ (rn := xn · x−1n+1, r-wt 1) : n ≥ 1.

Because all the relator weights are 1, this weighted normal closure equals theusual normal closureN = NΩ(X,wt)(R) of the relator set R. The cosets (ω, x)·Nin this quotient group Π(PHA) are called traces of words (ω, x) for the weightedalphabet (X,wt) and two such words (ω, x) and (ω′, x′) determine the sametrace if and only if they become similar when finitely many of the generatorsx1, x2, . . . , xn, . . . are identified with one another. The feature that (X,wt) isa weighted basis for Ω(X,wt) is lost in the formation of the quotient group;the cosets x · N for 1X 6= x ∈ X are all the same trace and do not generateΠ(PHA) in any sense. Even though all the members of the weighted basisbecome equal in the presented group Π(PHA), this group is not cyclic. Thisstrange feature is part of the fact Π(PHA) is not an omega-group: order-typeproducts are not well-defined in this quotient group ([S97]).

…

Figure 7: HarmonicArchipelago

The metric realization HA of PHA can be produced from the disc of lunarcrescents between the consecutive rings of the planar copy of the Hawaiian

3 EXAMPLES 27

earring by raising an interior portion of each lunar crescent to form a mountainof unit elevation. The result is called the harmonic archipelago; see Figure7. The loops in the Hawaiian earring correspond to words (ω, x) for theweighted generator set (X,wt) and path homotopies of these loops correspondto similarities modulo the weighted relator set (R, r-wt) of these words. Thenon-triviality of the presented group is easily visualized since, for compactnessconsiderations, any loop deforms continuously over at most finitely many ofthe mountainous crescents bounded by the consecutive rings.

Notice that the disc of cresents in Example 3.1 and the harmonic archipelagoin Example 3.3 have identical cellular decompositions and even isometric 1-skeletons. The great distinction between their fundamental groups π1(DC) =1 and π1(HA) is due to the different metric limiting behavior of their 2-cellsdemanded by relator weights of PDC and PHA.

Example 3.4 The harmonic projective plane on the Hawaiian earring.

The weighted presentation PHP :

< (xn, wt 1n) : n ≥ 1 : (rn := x2

n, r-wt1n) : n ≥ 1 >

presents the quotient group Ω(X,wt)/N(XX,wt) of the free omega-groupΩ(X,wt) on the same weighted basis (X,wt), modulo the weighted normalclosure N(XX,wt) of the weighted relator set of squares xx of members x ∈ X:

(XX,wt) = (1R, r-wt 0) ∪ (rn := x2n, r-wt

1n) : n ≥ 1

The metric realization HP of this weighted presentation can be created by theidentification of antipodal points in each 2-sphere in the surface of revolutionobtained by rotating the Hawaiian earring about its axis of symmetry. Itis a metric 1-point union of copies of the real projective plane of vanishingdiameters. The fundamental group of HP is seen in [B-S97(2)] to be anomega-group by use of [S97]; it can be viewed as a weighted free product[B-S97(2)] of copies of the finite cyclic group of order 2.

Example 3.5 The harmonic torus on the Hawaiian earring.

The weighted presentation PHT :

< (xn, wt 1n) : n ≥ 1 : (rmn := [xm, xn], r-wt 1

n) : m > n ≥ 1 >

presents the non-trivial quotient group Ω(X,wt)/N(R, r-wt) of the free omega-group Ω(X,wt) on the same weighted basis (X,wt) modulo the weighted nor-mal closure N(R, r-wt) of the weighted relator set

(R, r-wt) = (1R, r-wt 0) ∪ (rmn := [xm, xn], r-wt 1n) : m > n ≥ 1

3 EXAMPLES 28

of commutators [x, y] = xyx−1y−1 of elements of X.

The metric realization HT of this weighted presentation is a metric union ofcopies of the torus of vanishing diameters. The fundamental group of HT isseen in [B-S97(2)] to be an omega-group by use of [S97]. This omega-groupis nonabelian in spite of the presence of a set of pairwise commuting omega-generators. For example, the element x1 fails to commute with the harmonicproduct x2 · x3 · . . . · xk · . . . in the fundamental group π1(HT ). To see this,consider any similarity square

S : x1 · (x2 · x3 · . . . · xk · . . .) s∼ (x2 · x3 · . . . · xk · . . .) · x1 (mod R).

We may draw each relator disc as a square with oppositely oriented oppositesides having the same label. In the similarity square S, an x1-ribbon is formedby a tubular neighborhood of x1-arcs and is decorated by a finite number ofrelator discs for relators of the form rm1 := [xm, x1], m > 1. See Figure 8.

x2…x1 x3 x4

…x3 x4x2 x1

xjxj

x k

x k x l

x l

Figure 8: Similarity Square for PHT

The two appearances of the generator x1 in the labels on the opposite sides ofS must be joined by one such x1-ribbon. This x1-ribbon separates S into twopieces, either of which can be interpreted as a similarity square

S ′ : ws∼ x2 · x3 · . . . · xk · . . . (mod R)

where w is a finite word supported by the relator discs in the separating x1-ribbon. We note that no such similiarity square can exist. For there is an indexk ≥ 2 that does not appear in the finite word w, and so the xk-ribbon thatbegins at the lone boundary label xk in S ′ has nowhere to end, an impossibility.

The group π1(HT ) does contain an ordinary free abelian subgroup of count-ably infinite rank. Since the elements xn, n ≥ 1 commute pairwise in π1(HT ),the subgroup of finite words in the xn, n ≥ 1 in π1(HT ) is a homomorphic

3 EXAMPLES 29

image of the ordinary free abelian group⊕∞

n=1 Z with basis xn : n ≥ 1. Asimilarity argument using ribbons can be used to show that this homomor-phism is injective.

Example 3.6 The projective telescope PT through the Hawaiian earring.

The weighted presentation PPT :

< (xn, wt 1n) : n ≥ 1 : (rn := x−1

n+1 · x2n, r-wt

1n) : n ≥ 1 >

presents the non-trivial quotient group Ω(X,wt)/N(R, r-wt) of the free omega-group Ω(X,wt) on the same weighted basis (X,wt) modulo the weighted nor-mal closure N(R, r-wt) of the weighted relator set

(R, r-wt) = (1R, r-wt 0) ∪ (rn := x−1n+1 · x2

n, r-wt1n) : n ≥ 1.

Thus the group elements are cosets (ω, x) ·N(R, r-wt) of reduced order-typewords (ω, x) for the weighted generating set (X,wt); two reduced words (ω, x)and (ω′, x′) represent the same coset if and only if they are similar modulofinitely many applications of each relation xn+1 = x2

n, n ≥ 1. This quotientgroup is an omega-group; see [B-S97(2)].

The metric realization PPT can be viewed as the projective telescope PT ob-tained by an iteration of the procedure of replacing a disc dn+1 of diameter 1

n+1

in a projective plane Pn of diameter 1n

by a projective plane Pn+1 of diameter1

n+1, beginning with a projective plane P1 of diameter 1.

The fundamental group π1(PT ) is a non-trivial group. For example, for anyexponents εk = ±1 and indices m1 < m2 < . . . < mk < . . ., the word

xε1m1· xε2m2

· . . . · xεkmk · . . .

is not trivial in π1(PT ). To see this, consider any similarity square

S : xε1m1· xε2m2

· . . . · xεkmk · . . .s∼ 1X (mod R).

We may draw each relator disc for the relator rn := x−1n+1 · x2

n as a triangle,and we may interpret it as a place where an xn+1-ribbon (i.e., the tubularneighborhood of an xn+1-arc) dead-ends and an xn-ribbon (i.e., the tubularneighborhood of an xn-arc) is twisted. All xj-ribbons for the indices j < m1

can be eliminated from S. First of all, when j = 1 < m1, any x1-ribbon avoidsthe boundary, never dead-ends, and so closes up after an even number of twistsat appearances of r1-relator discs, since the generator x1 appears only in thefirst relator r1 := x−1

2 ·x21. Consecutive pairs of the r1-relator discs encountered

3 EXAMPLES 30

by the x1-ribbon can be discarded and the corresponding dead-ends of the x2-ribbons can be spliced together, so that the entire x1-ribbon can be deleted.Then the same argument applies inductively for all 1 < j < m1. But then thexm1-ribbon beginning at the first label can have no end, an impossibility.

The fundamental group π1(PT ) presented by PPT is non-abelian group eventhough the generators xn+1 become successive squares xn+1 = x2

n of one an-other in π1(PT ). For example, the loop that traverses all the even-indexedgenerating loops x2n (n ≥ 1), in succession, does not commute up to path-homotopy with the generating loop x1. In an assumed similarity square

S : x1 · (x2 · x4 · x6 · . . .) s∼ (x2 · x4 · x6 · . . .) · x1 (mod R),

the two appearances of the generator x1 in the labels on the opposite sides ofS must be joined by an x1-ribbon twisted at an even number of r1- relatortriangles, since the generator x1 appears only in the first relator r1 := x−1

2 ·x21.

See the similarity square in Figure 9. This x1-ribbon separates the similaritysquare S into pieces, each of which can be viewed as a similarity square

S ′ : 1Xs∼ xk2 · x4 · x6 · . . . (mod R),

But the product xk2 ·x4 ·x6 ·. . . cannot determine the identity in the group. Thisfollows via a case-by-case examination of the values k = 0,±1,±2, . . . and theprevious investigation of non-triviality in the group. Our non-commutativityconclusion is not obvious: modulo (R, r-wt), the word x1 does commutes withthe harmonic product x2x3 . . . xn . . ., which is similar modulo (R, r-wt) to x−2

1 .

x2…x1 x4 x6

…x4 x6x2 x1

x2

x2

x2

x2

Figure 9: Similarity Square for PPT

REFERENCES 31

References

[B-S97(1)] W. A. Bogley and A. J. Sieradski, Universal Path Spaces, submitted forpublication.

[B-S97(2)] W. A. Bogley and A. J. Sieradski, Omega-groups II: Weighted presentationsfor omega-groups, preprint.

[B-S97(3)] W A. Bogley and A. J. Sieradski, Omega groups III: Inverse limits andprofinite groups, preprint.

[C-C97(1)] J. W. Cannon and G. R. Conner, The combinatorial structure of the Hawai-ian earring group, preprint.

[C-C97(2)] J. W. Cannon and G. R. Conner, On the fundamental groups of one-dimensional spaces, preprint.

[C-C97(3)] J. W. Cannon and G. R. Conner, The big fundamental group, big Hawaiianearrings, and the big free groups, preprint.

[dS92] B. de Smit, The fundamental group of the Hawaiian earring is not free,Internat. J. Algebra Comp., 2 (1992) 33-37.

[G56] H. B. Griffiths, Infinite products of semigroups and local connectivity, Proc.London Math. Soc. (3) 6 (1956) 455-485.

[M-M86] J. W. Morgan and I. Morrison, A van Kampen theorem for weak joins,Proc. London Math. Soc. (3) 53 (1986) 562-576.

[S93] A. J. Sieradski, Algebraic topology for two-complexes. Chapter 2 in:Two-dimensional Homotopy and Combinatorial Group Theory (C. Hog-Angeloni, W. Metzler, and A. J. Sieradski, editors), London Math. Soc.Lecture Note Series 197 (1993), 51-96.

[S97] A. J. Sieradski, Omega-groups, preprint.

William A. BogleyDepartment of MathematicsKidder 368Oregon State UniversityCorvallis, OR [email protected]

Allan J. SieradskiDepartment of MathematicsUniversity of OregonEugene, OR [email protected]

weighted combinatorial group theory and wild metric complexes w. a. bogley oregon state university...

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