the r property for crystallographic groups of solmaths.sogang.ac.kr/jlee/symp-n/nims_jbl.pdf ·...

The R∞ propertyfor Crystallographic Groups of Sol

Jong Bum Lee1

(joint with Ku Yong Ha)

1Sogang University, Seoul, KOREA

Nielsen Theory and Related Topics 2013June 29, 2013

Jong Bum Lee (joint with Ku Yong Ha) The R∞ property for Crystallographic Groups of Sol

Outline

1 Reidemeister number and R∞ property2 Aim3 What is Sol?4 Solvable space forms problem5 Lattices of Sol6 Classification of space forms problem modeled on Sol7 Study of the R∞ property of space form groups modeled

on Sol


Def: Reidemeister number and R∞ Property

Let G be a group and let ϕ be an endomorphism of G.

The Reidemeister relation ∼ on G is defined as follows:

x ∼ y ⇔ ∃g ∈ G : y = gxϕ(g)−1.

The Reidemeister number R(ϕ) of ϕ is the cardinality of theReidemeister classes.

When R(ϕ) = ∞ for all automorphisms ϕ on G, we say that Ghas the R∞ property.


Aim

Our aim is to study the Reidemeister numbers for allautomorphisms on crystallographic groups of Sol.

We will show that most of the crystallographic groups of Solhave the R∞ property, extending Goncalves-Wong’s result forthe Bieberbach groups.

Remark: The work of discovering which groups have the R∞property was begun by A. Fel’shtyn and R. Hill.


The Lie group Sol

Sol = R2 oσ R where t ∈ R acts on R2 via the map

σ(t) =[et 00 e−t

].

Sol can be imbedded into Aff (R3) aset 0 0 x0 e−t 0 y0 0 1 t0 0 0 1

.


Space Forms Problem

According to Thurston, there are 8 kinds of geometries indimension 3.

R3, H3, S3, S2 × R, H2 × R, ˜SL(2, R), Nil , Sol.

A question naturally arisen is the problem of the classification ofclosed 3-orbifolds with a geometric structure modeled on one ofthese eight types, called the space forms problem.


Euclidean Space Forms Problem

The 3-dim Euclidean space forms problem

Aff (R3) = R3 o GL(3, R)O(3) is a maximal compact subgroup of GL(3, R)E(3) = R3 o O(3) = Isom(R3)

Discrete cocompact Π ⊂ E(3), called crystallographic groups.

Torsion-free discrete cocompact Π ⊂ E(3), called Bieberbachgroups.


Nilpotent Space Forms Problem

The 3-dim nilpotent space forms problem:

Aff (Nil) = Nil o Aut(Nil)O(2) is a maximal compact subgroup of Aut(Nil)Nil o O(2) = Isom(Nil)

Classify the (torsion-free) discrete cocompact Π ⊂ Isom(Nil):

Dekimpe, Igodt, Kim, KB Lee, Affine structures for closed3-dimensional manifolds with Nil -geometry, Quart. J. Math.Oxford Ser. (2), 46 (1995), 141–167.


Solvable Space forms Problem

The 3-dim solvable space forms problem:

Aff (Sol) = Solo Aut(Sol)D(4), the dihedral group of order 8, is a maximal compactsubgroup of Aut(Sol)Solo D(4) =∗ Isom(Sol)

Classify the (torsion-free) discrete cocompact Π ⊂ Isom(Sol):

K. Y. Ha and J. B. Lee, Crystallographic groups of Sol, Math.Nachr., 1–54 (2013)/ DOI 10.1002/mana.201200304.


Isom(Sol)

There are two non-equivariant left invariant Riemannian metricson Sol, and for those metrics the full isometry groups Isom(Sol)are isomorphic to Solo (Z2)2 ⊂ Solo D(4).

K. Y. Ha and J. B. Lee, Left invariant metrics and curvatures onsimply connected three-dimensional Lie groups, Math. Nachr.,282 (2009), 868–898.

K. Y. Ha and J. B. Lee, The isometry groups of simply connected3-diemnsional unimodular Lie groups, J. Geom. Phys., 62 (2012),no. 2, 189–203.


Associated to Isom(Sol) = Solo D(4), there is an exactcommutative diagram

1 1y yR2 =−−−−→ R2y y

1 −−−−→ Sol −−−−→ Isom(Sol) −−−−→ D(4) −−−−→ 1y y y=1 −−−−→ R −−−−→ R o D(4) −−−−→ D(4) −−−−→ 1y y

1 1


LetΠ ⊂ Isom(Sol) = Solo D(4), a crystallographic groupΓ = Π ∩ Sol, a lattice of SolΦ = Π/Γ, the holonomy group of Π

1 1y yZ2 =−−−−→ Z2y y

1 −−−−→ Γ −−−−→ Π −−−−→ Φ −−−−→ 1y y y=1 −−−−→ Z j−−−−→ Q π−−−−→ Φ −−−−→ 1y y

1 1


Lattices Γ of Sol

Let Γ be a lattice (i.e., a discrete cocompact subgroup) of

Sol = R2 oσ R.

Then the following diagram of short exact sequences iscommutative

1 −−−−→ R2 −−−−→ Sol −−−−→ R −−−−→ 1x x x1 −−−−→ Z2 −−−−→ Γ −−−−→ Z −−−−→ 1


Lattices of Sol

The following are mainly from

J. B. Lee and X. Zhao, Nielsen type numbers and homotopyminimal periods for maps on the 3-solvmanifolds, Algebr. Geom.Topol., 8 (2008), 563–580.

Given a lattice Γ of Sol, there is a hyperbolic integer matrix A

A =[`11 `12`21 `22

]such that

Γ = ΓA =〈

a1, a2, t | [a1, a2] = 1, tai t−1 = A(ai)〉

.

Notice also that `12 and `21 are nonzero and

PAP−1 = σ(t) =[et 00 e−t

]⇒ A ∈ SL(2, Z) with tr A > 2.


Crystallographic Groups of Sol

There are 9 kinds of crystallographic groups of Sol:

ΓA, Π1(k), Π±2 , Π3(k, k

′), Π4(k),

Π5(m, k, k′, n), Π6(k, k′), Π7(k), Π8(k, m)

and 4 kinds of Bieberbach groups of Sol:

ΓA, Π±2 , Π3(k, k

′), Π6(k, k′)

for a particular choice of k and k′.

For example,(1) ΓA = 〈a1, a2, t | [a1, a2] = 1, tai t−1 = A(ai)〉, a lattice of Sol(2) Π±2 = 〈a1, a2, s | [a1, a2] = 1, sais

−1 = N±(ai)〉

N± = −

[`11±1√

`11+`22±2`12√

`11+`22±2`21√

`11+`22±2`22±1√

`11+`22±2

](N2 = A and so [Π±2 : ΓA] = 2.)


(3)

Π3(k, k′) =

〈a1, a2, t , β

[a1, a2] = 1, tai t−1 = A(ai),βaiβ−1 = M(ai),β2 = ak , βtβ−1 = ak

′t−1

〉,

where

M =[−1 m

0 1

]and MAM−1 = A−1, and

(k, k′ − k) ∈ ker(I −M)im (I + M)

⊕ ker(A−M)im (A−1 + M)

.

Note that Π3(k, k′) is a Bieberbach group if and only ifm = 0, k = e2 and k′ − e2 6= 0. Further, [Π3(k, k′) : ΓA] = 2.


(4)

Π6(k, k′) =

〈a1, a2, α, β

[a1, a2] = 1,αaiα−1 = N+(ai), βaiβ−1 = M(ai),β2 = ak , βαβ−1 = ak

′α−1

〉,

where M is traceless with determinant −1 and MAM−1 = A−1,and

(k, k′ − k) ∈ ker(I −M)im (I + M)

⊕ ker(N −M)im (N−1 + M)

.

Note that Π6(k, k′) is a Bieberbach group if and only ifm = 0, k = e2 and k′ − e2 6= 0.Further, [Π3(k, k′) : ΓA] = 4.


On Bieberbach groups

A few remarks on the Bieberbach groups of Sol:

1 The quotient spaces ΓA\Sol and Π±2 \Sol are torus bundlesover the circle.

2 The quotient spaces Π3\Sol and Π6\Sol are the union oftwo twisted I-bundles over the Klein bottle, called thesapphire spaces.


Suppose there is a commutative diagram of groups:

1 −−−−→ Γ −−−−→ Π −−−−→ Π/Γ −−−−→ 1yϕ′ yϕ yϕ̄1 −−−−→ Γ −−−−→ Π −−−−→ Π/Γ −−−−→ 1

where the sequence is exact and where the quotient group Π/Γis finite.

Then for each α ∈ Π, we denote by τα the conjugation by α;then

1 −−−−→ Γ −−−−→ Π −−−−→ Π/Γ −−−−→ 1yταϕ′ yταϕ yτᾱϕ̄1 −−−−→ Γ −−−−→ Π −−−−→ Π/Γ −−−−→ 1


Then the following known fact is our main tool.

Theorem

When the previous commutative diagram is given, we have:

(1) R(ϕ) is finite if and only if R(ταϕ′) is finite for every α ∈ Π.(2) We have

R(ϕ) ≥ 1[Π : Γ]

∑ᾱ∈Π/Γ

R(ταϕ′).

When either side of the inequality is finite, then equalityoccurs if and only if fix(ταϕ) ⊂ Γ for each α ∈ Π.


To use the previous theorem, we need to find a (maximal)characteristic subgroup Γ of each crystallographic group Π.

To find such a specific characteristic subgroup, we will use thefollowing Lemmas:

Lemma (JBL-KBL)

Let S be simply connected solvable Lie groups, and letΠ ⊂ Aff (S) be a finite extension of a lattice Γ of S. Then thereexists a fully invariant subgroup Λ ⊂ Γ of Π, which is of finiteindex.

Lemma (JBL-Zhao)

Let Λ be a lattice of Sol = R2 oσ R. Then Λ ∩ R2 ∼= Z2 is a fullyinvariant subgroup of Λ.


Conclusion

For every automorphism ϕ,

ΓA R(ϕ) = 4,∞Π1(k) R(ϕ) = 4, 8,∞Π+2 R(ϕ) = 4,∞

Π−2Π3(k, k′)Π4(k)Π5(m, k, k′, n) R(ϕ) = ∞Π6(k, k′)Π7(k)Π8(k, m)

This extends Goncalves-Wong’s result for the Bieberbachgroups.


We show for which automorphisms ϕ, R(ϕ) = 4 or 8.

Let Π = ΓA,Π1(k) or Π+2 .

Let ϕ : Π → Π be an automorphism.Then ΓA is a characteristic subgroup of Π.So, ϕ induces an automorphism ϕ′ on ΓA = 〈aa, a2, t〉, which isof the form

ϕ(ai) = am1i1 a

m2i2 , ϕ(t) = a

p1a

q2 t

r

with r = ±1.

Then det[mij ] = 1 and r = −1 if and only if R(ϕ) = 4 or 8.We can also give precise conditions on which R(ϕ) = 8.


Many many thanks!

See all of you at the next Nielsen conference.


the r property for crystallographic groups of solmaths.sogang.ac.kr/jlee/symp-n/nims_jbl.pdf ·...

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