NEW ZEALAND JOURNAL OF MATHEMATICS Volume 24 (1995), 29-40

F L A T 4 -M A N IF O L D G R O U P S

J o n a t h a n A. H i l l m a n

(Received February 1994)

Abstract. We com pute the fundamental groups of flat 4-manifolds ab initio, using direct and elem entary arguments.

1. In tro d u c tio nA group 7r is a flat n-manifold group if it is torsion free and has a normal subgroup

of finite index which is isomorphic to Z n. (These are necessary and sufficient con­ditions for 7r to be the fundamental group of a closed flat riemannian n-manifold). The action of 7r by conjugation on the maximal abelian normal subgroup A ( tt) induces a faithful action of tt/A ( tt) on A(n). On choosing an isomorphism of A (k ) with Z n we may identify 7r/A (7r) with a subgroup of G L ( n ,Z ); this subgroup is called the holonomy group of 7r, and is well defined up to conjugacy in G L (n ,Z ) . We say th a t 7r is orientable if the holonomy group lies in S L (n ,Z ) . (The group is orientable if and only if the corresponding closed flat riemannian 4-manifold is orientable).

There have been several independent computations of the flat 4-manifold groups. The consensus of the references [Ca,CS,Le] is th a t there are 27 orientable groups and 48 nonorientable groups. (See page 126 of [Wo]). However the tables of 4-dimensional crystallographic groups in [B] list only 74 torsion free groups. As the earlier computations were never published in detail, and the computer-generated tables of [B] give little insight into how these groups arise, we present our own computations here. The present approach is motivated by the observation tha t every closed 4-dimensional infrasolvmanifold is either a mapping torus or is the union of two twisted /-bundles over 3-dimensional infrasolvmanifolds. (See the Corollary to the Lemma in §3 below). The arguments are direct and elementary. However comparison of our preliminary results with the tables of [B] suggested tha t there were two redundancies in the list of groups with finite abelianization, and we have relied upon computational algebra to eliminate these redundancies. Our conclusions as to the numbers of groups with abelianization of given rank, isomorphism type of holonomy group and orientation type agree with those of [B]. (We have not attem pted to make the lists correspond).

I would like to thank John Cannon for his advice, Derek Holt for finding the final two isomorphisms and Eamonn O’Brien for confirming th a t there are no others.

2 . F la t 3 -M an ifo ld G ro u p sIf G is a group then 1(G) = {g G G | 3n > 0,gn G G'} is a characteristic

subgroup of G , and the quotient G /I ( G ) is a torsion free abelian group of rank Pi(G). We shall organize our determination of the flat 4-manifold groups 7r in terms of I(ir). Let D = (Z /2Z ) * (Z / 2 Z ) be the infinite dihedral group.

1991 A M S Mathematics Subject Classification: Primary 57N13. Keywords and Phrases: flat 4-manifold, infrasolvmanifold.

30 JONATHAN A. HILLMAN

The flat n-manifold groups for n < 2 are Z, Z 2 and K — Z x _i Z, the Klein bottle group. There are six orientable and four nonorientable flat 3-manifold groups. (See pages 117-121 of [Wo]). The first of the orientable flat 3-manifold groups G\ - Gq is G\ — Z 3. The next four have I (G i) = Z 2 and are semidirect products

Z 2 x T Z where T — — / , ~ j ^ ^ ® ~ o ) ° r ( l _ 1 ) ’ resPectively> *s an element of finite order in S L (2 ,Z ) . (See pages 82-85 of [Zg] for the finite subgroups of G L{2,Z)) . These groups all have cyclic holonomy groups, of orders 2 ,3 ,4 and 6, respectively. The group Gq is the group of the Hantzsche-Wendt flat 3-manifold, and has a presentation (x ,y \ x y 2x ~1 = y~ 2 , y x 2y ~ 1 = x ~ 2). Its maximal abelian normal subgroup is generated by x2,y 2 and (xy )2 and its holonomy group is the diagonal subgroup of SX(3, Z), which is isomorphic to (Z /2 Z )2. (This group is the generalized free product of two copies of K , amalgamated over their maximal abelian subgroups, and so maps onto D).

The nonorientable flat 3-manifold groups B\ - B 4 are semidirect products K x q Z , corresponding to the classes in Out(AT) = (Z /2 Z )2. In terms of the presentation (.x ,y | x y x ~ l — y ~ l ) for K these classes are represented by the automorphisms 9 which fix y and send x to x , x y , x ~ l and x ~ 1y , respectively. The groups B\ and £?2 are also semidirect products Z 2 x j Z, where T = or ^ g j has

determinant —1 and T 2 — I. They have holonomy groups of order 2, while the holonomy groups of B 3 and B 4 are isomorphic to (Z /2 Z )2.

All the flat 3-manifold groups either map onto Z or map onto D. The methods of this paper may be easily adapted to find all such groups. Assuming these are all known we may use Sylow theory and a little topology to show th a t there are no others. We sketch here such an argument. Suppose th a t tt is a flat 3-manifold group with finite abelianization. Then 0 = x ( 7 r ) = 1 + ^ ( t t jQ ) — s o

/?3 (7r;Q) 7 0 and n must be orientable. Hence the holonomy group F = 7r/A(ir) is a subgroup of SL (3 ,Z ) . Let / be a nontrivial element of F. Then / has order 2 ,3 ,4 or 6, and has a +l-eigenspace of rank 1 , since it is orientation preserving. This eigenspace is invariant under the action of the normalizer N p {( f ) ) , and it is easily seen th a t the induced action of N p ( ( f ) ) on the quotient space is faithful. (See Lemma VI.2 of [HI]). Thus N p ( ( f ) ) is isomorphic to a subgroup of G L(2, Z) and so is cyclic or dihedral of order dividing 24. Since p-groups have nontrivial centres it follows th a t the Sylow 2- and 3-subgroups of F have order dividing 8 and3 respectively. Hence the order of F divides 24. If F has a nontrivial cyclic normal subgroup then 7r has a normal subgroup isomorphic to Z 2 and hence maps onto Z or D. Otherwise F has a nontrivial Sylow 3-subgroup which is not normal in F. The number of Sylow 3-subgroups is congruent to 1 m od(3) and divides the order of F, and the natural perm utation representation of F on the set of such subgroups is faithful. Hence F must be A 4 or S 4 , and so contains V = (Z /2 Z )2 as a normal subgroup. But any orientable flat 3-manifold group with holonomy V must have finite abelianization. As Z /3 Z cannot act freely on a Q-homology 3-sphere (by the Lefshetz fixed point theorem) it follows th a t A 4 cannot be the holonomy group of a flat 3-manifold. Hence we may exclude S4 also.

FLAT 4-MANIFOLD GROUPS 31

3. A L em m a o n 4 -M an ifo ld s w ith E u le r C h a ra c te r is t ic 0The following simple application of Poincare duality shall enable us to reduce the

classification of flat 4-manifold groups to questions about flat 3-manifold groups.

L em m a. Let M be a PD^-complex with x{M ) = 0. I f M is orientable then0 and so tt — 7Ti (M ) maps onto Z. I f H 1 (M; Z) = 0 then it maps

onto D.

P ro o f . The covering space M w corresponding to W = J(er(w i(M )) is orientable and = 2 — 2/3\ (M w ) + @2{Mw) — [tt : W ]x(M ) = 0. Therefore0 i (W ) — (3i(Mw) > 0 and so W /I ( W ) = Z r for some r > 0. Since I (W ) is characteristic in W it is normal in tt. As [7r : W] < 2 it follows easily th a t tt/ I ( W ) maps onto Z or D. |

C o ro lla ry . I f ir is torsion free and virtually poly-Z of Hirsch length 4 then it is the fundamental group of a closed smooth 4-manifold M which is either a mapping torus of a self homeomorphism of a closed 3-dimensional infrasolvmanifold or is the union of two twisted I-bundles over such a 3-manifold.

P ro o f . The Eilenberg-Mac Lane space K ( 7r ,l) is a PZ^-complex with Euler characteristic 0. By the lemma, either there is an epimorphism (j) : 7r —* Z, in which case tt is a semidirect product G Xq Z where G = ker(0), or tt = G\ *G G 2 and [Gi : G] = [G2 : G\ = 2. The subgroups G, G 1 and G2 are torsion free and virtually poly-Z. Since in each case 7t /G has Hirsch length 1 these subgroups have Hirsch length 3 and so are fundamental groups of closed 3-dimensional infrasolv- manifolds. The corollary now follows by standard 3-manifold topology. |

The Corollary may be stated and proven in purely algebraic terms, using the fact th a t torsion free virtually poly-Z groups are Poincare Duality groups. (See Chapter III of [Bi]).

Using Lie theory, it can be shown th a t groups 7r as in the Corollary are fundamen­tal groups of closed 4-dimensional infrasolvmanifolds [AJ], and using 4-dimensional surgery it can be shown that aspherical closed 4-manifolds with such fundamental groups are determined up to homeomorphism by their fundamental group and the fact th a t they have Euler characteristic 0 (see Chapter VI of [H2]).

4. A u to m o rp h ism s o f F la t 3 -M an ifo ld G ro u p sIf 7r is a flat 4-manifold group with infinite abelianization -k/'k' then 7r is a semidi­

rect product G Xq Z where G is a flat 3-manifold group. Such semidirect products are determined up to isomorphism by the class of the automorphism 9 in the outer automorphism group Out(G). (More precisely, if the images of 9 and 9 in Out(G) are conjugate up to inversion then the corresponding semidirect products are isomorphic). Thus an essential step in finding the flat 4-manifold groups is to determine the outer automorphism groups of the flat 3-manifold groups.

Clearly O ut(G i) = A ut(G i) = G L(3 ,Z). If 2 < i < 5 let t 6 Gi repre­sent a generator of the quotient G i/I(G i) = Z. The automorphisms of Gi must preserve the characteristic subgroup I{Gi) and so may be identified with triples (v ,A ,£) € Z 2 x G L (2 ,Z ) x Z x such th a t A T A ~ l — T £ and which act via A

32 JONATHAN A. HILLMAN

on I (G i ) = Z 2 and send t to tev. Such an automorphism is orientation preserv­ing if and only if e = det(^4). The multiplication is given by (v, A ,e)(w , B,r]) = (Si> + A w ,A B ,e r j) : where S = I if r] = 1 and E = —T e if 77 — —1. The inner automorphisms are generated by (0,T , 1) and ((T — I ) Z 2 , I , l) .

In particular, Aut(C?2) = (Z 2 x Q G L(2, Z )) x Z x , where a is the natural action of G L(2 ,Z ) on Z 2, for E is always / if T = —I. The involution (0 ,/, —1) is central in Aut(C?2), and is orientation reversing. Hence O ut(G 2) — { (Z /2 Z )2 Xpa PGL(2, Z)) x (Z / 2 Z ), where P a is the induced action of P G L (2 ,Z ) on (Z /2 Z )2.

If n — 3 ,4 or 5 the normal subgroup I(G{) may be viewed as a module over the ring R = Z[t]/(f)(t), where 4>{t) = t 2 + t + l , t 2 + l or t 2 — t + 1, respectively. As these rings are principal ideal domains and I(Gi) is torsion free of rank 2 as an abelian group, in each case it is free of rank 1 as an i?-module. Thus matrices A such th a t A T = T A correspond to units of R. Hence automorphisms of Gi which induce the identity on G i/ I (G i ) have the form ( v , ± T m, 1), for some m G Z andv € Z 2. There is also an involution (0, q J , —l) which sends t to t~ l . In all

cases e = det(^l). It follows th a t Out(Ga) = S3 x (Z /2Z ) , Out(G<4) = (Z /2 Z )2 and Out(Gs) = Z j 2Z. All these automorphisms are orientation preserving.

Our description of Out(Gg) is taken from page 98 of [HI]. (See [Zn] for an alternative description). Let a, 6, c, e ,i and j be the automorphisms of Gq which send x to x _1, x, x, y2x, y, xy and y to y, y -1 , (xy)2y, y, x, x, respectively. Then Out(G*6) has a presentation

(a, 6, c, e, i, j \ a2 = b2 — c2 — e2 — i2 — j 6 = 1 , a, 6, c, e commute, iai = b,

i d = ae, j a j ~ l = c, j b j ~ x = abc, j c j ~ l = be, j e j ~ l = &c, j 3 = abce, (j i )2 = be).

The generators a, 6, c, and j represent orientation reversing automorphisms.The group B\ — Z x K has a presentation

( t , x ,y | tx = xt, ty = y t , x y x ~ l = y ~ l ).

An automorphism of B \ must preserve the centre ( B 1 (which has basis t , x 2) and I (B 1) (which is generated by y). Thus the automorphisms of B\ may be identified with triples (A , m ,e) € r 2x Z x Z x , where I^ is the subgroup of GL{ 2, Z) consisting of matrices congruent mod(2) to upper triangular matrices. Such an automorphism sends t to tax b, x to tcx dym and y to ye, and induces multiplication by A on B \ / I { B \ ) = Z 2. Composition of automorphisms is given by (A ,m ,£ ) ( B ,n , i7) = (.A B , m + en, erf). The inner automorphisms are generated by ( /, 1, —1) and ( /, 2,1), and so O u t(S i) = I?2 x (Z / 2 Z ).

The group B 2 has a presentation

{ t ,x ,y | t x t~ x = x y , t y = y t , x y x ~ l = y_1).

Automorphisms of B 2 may be identified with triples (A , (m, n), e) € T2 x Z 2 x Z x , such th a t m = (A n — e)/2. Such an automorphism sends t to tax by rn1 x to tcx dyn and y to y£, and induces multiplication by A on B 2/ I ( B 2) = Z 2. The automor­phisms which induce the identity on B 2/ I ( B 2) are all inner, and so Out(£?2) = IV

FLAT 4-MANIFOLD GROUPS 33

The group B 3 has a presentation ( t , x ,y \ t x t~ l = x ~ l , ty — y t , x y x ~ l = y ~ l ). An automorphism of B 3 must preserve I(B s) = K (which is generated by x ,y ) and / ( / ( S 3)) (which is generated by y). It follows easily th a t Out(B^) = (Z /2 Z )3, and is generated by the classes of the automorphisms which fix y and send t to t~ 1 , t , t x 2 and x to x , x ~ x,x , respectively.

A similar argument using the presentation

( t , x ,y | t x t ~1 = x ~ l y , ty = y t , x y x ~ l = y -1 )

for B 4 shows th a t Out(i?4) = (Z /2Z )3, and is generated by the classes of the auto­morphisms which fix y and send t to t~ 1y ~ 1 , t : t x 2 and x to x , x _ 1 ,x , respectively.

5. F la t 4 -M an ifo ld G ro u p s w ith In fin ite A b e lia n iz a tio nLet 7r be a flat 4-manifold group, H = I{7r), (3 = /?i(7r) and h = h(H). Then

i t /H = and h + (3 = 4. If i f is abelian then Cn (H) is a nilpotent normal subgroup of 7r and so is a subgroup of the Hirsch-Plot kin radical y/rr, which is here the maximal abelian normal subgroup A(ir). Hence C^^H) = A (tt) and the holonomy group is isomorphic to 7r/C'7r( i/) .

h = 0. In this case H = 1, so tt = Z 4 and is orientable.h = 1. In this case H = Z and ix is nonabelian, so -k/C ^{H ) = Z /2 Z . Hence 7r

has a presentation of the form

(£, x, y, z | t x t~ l = x z a, t y t~ l = y z b, t z t~ l = 2_1, x , y , z commute),

for some integers a, b. On replacing x by xy or interchanging x and y if necessary we may assume that a is even. On then replacing x by x z a 2 and y by yz^b 2 we may assume th a t a = 0 and b = 0 or 1. Thus 7r is a semidirect product Z 3 x T Z, where the normal subgroup Z 3 is generated by the images of x, y and z, andthe action of t is determined by a m atrix T = in G L(3 ,Z ). Hence

t t ^ Z x B 1 = Z 2 x K or Z x S 2. Both of these groups are nonorientable.h = 2. If H = Z 2 and ir/Cn(H) is cyclic then we may again assume th a t n is a

semidirect product Z 3 x T Z, where T = ^ j j j , with ji = (£) and U € G L(2, Z) is

of order 2,3,4 or 6 and does not have 1 as an eigenvalue. Thus U = —I 2 , ( j — 1 ) ’

( l _ 0 ) or ( 1 I 1 ) ' 111 eac^ case c^°^ce a = = 0 leads to a group of the form 7r = Z x G, where G is an orientable flat 3-manifold group with (3\ (G) — 1. For each of the first three of these matrices there is one other possible group. (Note tha t

conjugating T by ^ replaces /x by /it + {I2 — U)u). However if U — ~ j then I 2 — U is invertible and so Z x G5 is the only possibility. All seven of these groups are orientable.

If H = Z 2 and k / C n {H) is not cyclic then ,k /C ^{H ) = [Z /2 Z )2. There are two conjugacy classes of embeddings of (Z /2 Z )2 in G L(2,Z). One has image the subgroup of diagonal matrices. The corresponding groups 7r have presentations of the form

(t, u , x , y | tx = x t , t y t~ Y = y -1 , u x u ~ x = x ~l , u y u ~ l = y ~ l ,xy = y x , tu = x mynu t),

3 4 JONATHAN A. HILLMAN

for some integers ra, n. On replacing t by ££_ [m/ 2]y[n/ 2] if necessary we may assume th a t 0 < ra, n < 1. On then replacing t by tu and interchanging x and y if necessary we may assume th a t m < n. The only infinite cyclic subgroups of H which are normal in 7r are the subgroups (x) and (y). On comparing the quotients of these groups 7r by such subgroups we see th a t the three possibilities are distinct. The other embedding of ( Z / 2 Z ) 2 in G L ( 2 , Z ) has image generated by — I and ^ J j .

The corresponding groups n have presentations of the form

( t ,u , x , y | t x t~ l — y , t y t~ l = x , u x u ~ x = x _1, uyu ~ x = y_1,

xy = y x , tu = x mynu t) :

for some integers m, n. On replacing t by t x Km -n)/2l and u by u x ~ m if necessary we may assume th a t m = 0 and n = 0 or 1. Thus there two such groups. All five of these groups are nonorientable.

Otherwise, H = K , 1(H) = Z and G — n /1 (H ) is a flat 3-manifold group with /3i(G) — 2, but with 1(G) = H / I ( H ) not contained in G' (since it acts nontrivially on 1(H)). Therefore G = B \ — Z x K , and so has a presentation ( t , x ,y | tx = x t , t y = y t , x y x ~ l = y ~ l ). If w : G —+ A ut(Z) is a homomorphism which restricts nontrivially to 1(G) then we may assume (up to isomorphism of G) th a t w(x) = 1 and w(y) — —1. Groups tv which are extensions of Z x K by Z

corresponding to the action with w(t) — w (= ± 1) have presentations of the form

(t, x , y , z | t x t~ l = x z a, t y t~ l = y z b, t z t~ l — z w, x y x ~ l = y ~ 1 z c,

x z = zx , y z y ~ l = z ~ l ).

Any group with such a presentation is easily seen to be an extension of Z x K by a cyclic normal subgroup. However conjugating the fourth relation leads to the equation

t x t~ l ty t~ l ( tx t~ l )~ l = t x y x ~ l t~ l = t y ~ l z ct ~1 = ty t~ x ( tz t~ x)c

which simplifies to x z ay z bz~ ax ~ 1 — (yzb)~ 1z wc and hence to z c~2a = z wc. Hence this cyclic normal subgroup is finite unless 2a = (1 — w)c.

Suppose first th a t w — 1. Then z 2a = 1 and so we must have a = 0. On replacing t by tz\h!2\ and x by xz^c^2\ if necessary, we may assume th a t 0 < b, c < 1. If 6 = 0 then 7r = Z x Bs or Z x 5 4. Otherwise, after further replacing x by tx z if necessary we may assume th a t c = 0. The three remaining possibilities may be distinguished by their abelianizations, and so there are three such groups. In each case the subgroup generated by { t , x 2 , y 2, z } is maximal abelian, and the holonomy group is isomorphic to ( Z / 2 Z ) 2 .

If instead w = — 1 then z 2(<c~a) = 1 and so we must have a = c. On replacing y by yz^h!2\ and x by xz^c!2 if necessary we may assume th a t 0 < b, c < 1. If 6 = 1 then after replacing x by txy, if necessary, we may assume th a t a = 0. I f a = 6 = 0 then 7r / 7r' = Z 2 © (Z /2 Z )2. The remaining two possibilities both have abelianization Z 2 ® (Z /2Z ) , but one has centre of rank 2 and the other has centre of rank 1. Thus there are three such groups. The subgroup generated by { ty , x 2, y2, z} is maximal

FLAT 4-MANIFOLD GROUPS 35

abelian, and the holonomy group is isomorphic to (Z /2 Z )2. All of these groups 7r with / ( 7r) = K are nonorientable.

h = 3. In this case 7r is uniquely a semidirect product n = H Xq Z, where H is a flat 3-manifold group and 9 is an automorphism of H such tha t the induced automorphism of H /I ( H ) has no eigenvalue 1, and whose image in O u t(H) has finite order. Two such automorphisms 9 and 9 determine isomorphic groups if and only if their images in O u t(H) are conjugate up to inversion.

Since A(G) is the maximal abelian normal subgroup of G it is normal in 7r. It follows easily tha t A(ir) flG = A(G). Hence the holonomy group of G is isomorphic to a normal subgroup of the holonomy subgroup of 7r, with quotient cyclic of order dividing the order of 9 in Out(G). (The order of the quotient can be strictly smaller).

If H = Z 3 then O u t(H) = G L(3 ,Z ). If T G G L(3 ,Z ) has finite order n and /3\ (Z 3 X t Z ) — 1 then either T = — I or n = 4 or 6 and the characteristic polynomial of T is (t + 1 )<f)(t) with (f)(t) = t2 + 1, t2 + t + 1 or t 2 — t + 1. In the latter cases T

is conjugate to a m atrix of the form ~ J where A — ^ ~o)> ( l — l ) or

~ J J , respectively. The row vector /i, = ( m i , m 2) is well defined mod Z 2(A + 1).

Thus there are seven such conjugacy classes. All but one pair (corresponding to

( l _ l ) anc ^ ^ + ^)) are self-inverse, and so there are six such groups.The holonomy group is cyclic, of order equal to the order of T. As such matrices all have determ inant —1 all of these groups are nonorientable.

If H = Gi for 2 < i < 5 the corresponding automorphism 9 — (v , A , e) must have e — —1, for otherwise /3i (7r) = 2. We have Out((?2) — ({Z /2 Z )2 xP G L (2 , Z)) x (Z /2Z) . The five conjugacy classes of finite order in P G L (2 ,Z ) are represented

by the matrices / , ^ _ o ) ’ ( l o ) ’ (0 — l ) anc ( —1 l ) ' num^ ersconjugacy classes in O ut(G 2) with e = — 1 corresponding to these matrices are two, two, two, three and one, respectively. All of these conjugacy classes are self­inverse. Of these, only the two conjugacy classes corresponding to j- j and the

three conjugacy classes corresponding to give rise to orientable groups.

The holonomy groups are all isomorphic to (Z /2 Z )2, except when A — ^ — or

| J , when they are isomorphic to Z /4 Z or Z /6 Z © Z /2 Z , respectively. There are five orientable groups and five nonorientable groups.

As Out((?3) = S3 x (Z /2Z ) , Out(C?4) = ( Z /2 Z ) 2 and O ut(G 5) = Z /2 Z , there are three, two and one conjugacy classes corresponding to automorphisms with e — — 1 , respectively, and all these conjugacy classes are closed under inversion. The holonomy groups are dihedral of order 6,8 and 12, respectively. The six such groups are all orientable.

The centre of Out(Ge) is generated by the image of ab, and the image of ce in the quotient O u t(Go)/(ab) generates a central Z /2 Z direct factor. The quotient O u t(Ge)/(ab,ce) is isomorphic to the semidirect product of a normal subgroup (Z /2 Z )2 (generated by the images of a and c) with S 3 (generated by the images of ia and j ) , and has five conjugacy classes, represented by

36 JONATHAN A. HILLMAN

1 , a , i , j and ci. Hence Out(C?6)/(afe) has ten conjugacy classes, represented by 1 ,ce ,a ,a ce , i ,ce i , j ,c e j ,c i and cice = ei. Thus Out(Ge) itself has between 10 and 20 conjugacy classes. In fact Out(G'e) has 14 conjugacy classes, of which those rep­resented by l ,ab ,ace,bce,i,cej,abcej and ei are orientation preserving, and those represented by a ,ce ,ce i , j ,ab j and ci are orientation reversing. All of these classes are self inverse, except for j and abj, which are mutually inverse ( j -1 = ai(abj)ia). The holonomy groups corresponding to the classes 1 ,ab,ace and bee are isomor­phic to (Z /2 Z )2, those corresponding to a and ce are isomorphic to (Z /2 Z )3, those corresponding to i , ei, cei and ci are dihedral of order 8, those corresponding to cej and abcej are isomorphic to A \ and the one corresponding to j has order 24. There are eight orientable groups and five nonorientable groups.

All the remaining cases give rise to nonorientable groups.H = Z x K . If a m atrix A in T2 has finite order then as its trace is even the

order must be 1,2 or 4. If moreover A does not have 1 as an eigenvalue then either A = — I or A has order 4 and is conjugate (in r 2) to } )• Each of the fourcorresponding conjugacy classes in T2 x Z x is self inverse, and so there are four such groups. The holonomy groups are isomorphic to Z /n Z ® Z /2Z , where n = 2 or 4 is the order of A.

H = B 2. As O u t(-82) — r 2 there are two relevant conjugacy classes and hence two such groups. The holonomy groups are again isomorphic to Z j n Z © Z /2Z , where n = 2 or 4 is the order of A.

H = B 3 or B 4. In each case O u t(H) = (Z /2 Z )3, and there are four outer automorphism classes determining semidirect products with [3 = 1 . (Note tha t here conjugacy classes are singletons and are self-inverse). The holonomy groups are all isomorphic to (Z /2 Z )3.

6 . F la t 4 -m an ifo ld G ro u p s w ith F in ite A b e lia n iz a tio nThere remains the case when 7r /7t ' is finite (equivalently, h = 4). By the Lemma

if 7r is such a flat 4-manifold group it is nonorientable and is isomorphic to a generalized free product J J , where (j) is an isomorphism from G < J to G < J and [J : G] = [J : G] = 2. The groups G, J and J are then flat 3-manifold groups. If A and A are automorphisms of G and G which extend to J and J , respectively, then J *<£ J and J *X4>\ J are isomorphic, and so we shall say th a t 0 and A0A are equivalent isomorphisms. The major difficulty in handling these cases is th a t some such flat 4-manifold groups split as a generalised free product in several essentially distinct ways.

It follows from the Mayer-Vietoris sequence th a t Hi(G\ Q) maps onto H \ ( J ; Q)(B H i(J ;Q ) , and hence th a t (3\(J) + (3\{J) < (3\(G). Since G3, G4, B 3 and 5 4 are only subgroups of other flat 3-manifold groups via maps inducing isomorphisms on

and G5 and Gq are not index 2 subgroups of any flat 3-manifold group we may assume th a t G = Z 3, (?2, B i or B 2 . If j and j are the automorphisms of A (J) and A (J ) determined by conjugation in J and J , respectively, then tv is a flat 4-manifold group if and only if $ = jA((j))~l jA(cj)) has finite order. In particular, the trace of $ must have absolute value at most 3. At this point detailed compu­tation seems unavoidable. (We note in passing tha t any generalised free product

FLAT 4-MANIFOLD GROUPS 3 7

J * G J with G = G3 , G4 , B 3 or B 4 , J and J torsion free and [J : G] = [J : G\ = 2 is a flat 4-manifold group, since Out(G) is then finite. However all such groups have infinite abelianization).

Suppose first th a t G = Z 3, with basis {x , y , z }. Then J and J must have holonomy of order < 2, and j3\(J) + f i i(J) < 3. Hence we may assume th a t J = G2

and J = G2, B\ or # 2- In each case we have G = A (J ) and G = A(J). We may assume th a t J and J are generated by G and elements s and t, respectively, such tha t s2 = x and t2 £ G. We may also assume th a t the action of s on G has m atrix j = with respect to the basis {x ,y ,z} . Fix an isomorphism tp : G —> G

and let T = A(<f>)~1 jA(<p) — be the m atrix corresponding to the action

of t on G. (Here 7 is a 2 x 1 column vector, 6 is a 1 x 2 row vector and D is a 2 x 2 matrix, possibly singular). Then T 2 — I and so the trace of T is odd. Since j = I mod(2) the trace of $ = j T is also odd, and so $ cannot have order 3 or 6. Therefore <$4 = I. If $ = I then 'k/'k' is infinite. If $ has order 2 then j T = T j and so 7 = 0, 8 = 0 and D 2 — 1 2- Moreover we must have a — —1 for otherwise 'k/'k' is infinite. After conjugating T by a m atrix commuting with j if necessary we may assume that D = I 2 or (Since J must be torsion free we cannot

have D = f ^ ^ ) ) . These two matrices correspond to the generalized free products G2 *4, B \ , with presentation

(s, 2, t | s i2s _1 = t~ 2, s z s ~ 1 = z ~ 1, ts 2t~ 1 = s ~ 2, tz = zt)

and G2 *(j> G2 , with presentation

{s ,z , t | s t2s _1 = t~ 2, s z s~ 1 — z ~ 1, ts2t~ l — s~2, t z t~ l = z ~ 1),

respectively. These groups each have holonomy group isomorphic to (Z /2 Z )2. If $ has order 4 then we must have ( j T )2 = (_j T ) ~ 2 = ( T j )2 and so ( j T ) 2 commutes with j . It can then be shown that after conjugating T by a m atrix commuting with j if necessary we may assume that T is the elementary m atrix which interchanges the first and third rows. The corresponding group G2 *<f> B 2 has a presentation

(s ,2, t | s t2s ~ 1 = t~2, s z s~ 1 = z ~ 1, t s 2t~ l = 2, t z t~ l — s2).

Its holonomy group is isomorphic to the dihedral group of order 8.If G = Bi or B 2 then J and J are nonorientable and /3\(J) + /3i(J) < 2. Hence

J and J are £3 or B 4 . Since neither of these groups contains B 2 as a subgroup we must have G = B \. In each case there are two essentially different embeddings of Bi as an index 2 subgroup of £3 or B±. (The image of one contains I(B i) while the other does not). In all cases we find th a t j and j are diagonal matrices with

determ inant —1 , and th a t A(4>) = ± l ) or some ^ e Calculation now

shows th a t if $ has finite order then M is diagonal and hence (3\ (J J) > 0. Thus there are no flat 4-manifold groups with finite abelianization which are generalized free products with amalgamation over copies of B\ or B 2 ■

38 JONATHAN A. HILLMAN

If G = G2 then (3\{J) + /3\(J) < 1, so we may assume th a t J = Gq. The other factor J must then be one of G2, G4 , Gq, or B 4, and then every amalgamation has finite abelianization. In each case the images of any two embeddings of G2 in one of these groups are equivalent up to composition with an automorphism of the larger group. In all cases the matrices for j and j have the form ^ ^ where

N A — I E G L(2 ,Z ) , and A{4>) = ^ ^ for some M E G L(2 ,Z ). Calculation shows th a t $ has finite order if and only if M is in the dihedral subgroup Dg of G L (2 ,Z ) generated by the diagonal matrices and ^ ^ q ^ . (In other words, eitherM is diagonal or both diagonal elements of M are 0). Now the subgroup of Aut(G?2) consisting of automorphisms which extend to Gq is (Z 2 x a Dg) x Z x . Hence any two such isomorphisms ip from G to G are equivalent, and so there is an unique such flat 4-manifold group G q ^^J for each of these choices of J. The corresponding presentations are

( x ,y ,u | x u x ~ l = y 2 = u2, y x 2y ~ 1 = x~2, u (x y )2 = (x y ) 2u ),

( x ,y ,u | y x 2y ~ l = x ~ 2, uy 2u ~ l = (xy)2, u(xy)2u ~ l = y ~2, x = u 2),

( x ,y ,u | x y 2x ~1 = y ~2, y x 2y ~ 1 = u x 2u ~ l = x ~ 2, y 2 — u2, yxy = uxu),

(x , y , t | x y 2x ~ l = y~ 2, y x 2y ~ l = x ~2, x 2 = t 2, y 2 = (t~ l x )2, t (xy ) 2 = (xy ) 2t )

and

( x ,y , t | x y 2x ~ 1 = y~2, y x 2y ~ l = x ~ 2, x 2 = t2 (xy)2,

y 2 = (1t~xx )2, t ( x y )2 = (xy ) 2t ),

respectively. The corresponding holonomy groups are isomorphic to (Z /2 Z )3, Dg, (Z /2 Z )2, ( Z /2 Z ) 3 and (Z /2 Z )3, respectively.

Thus we have found eight generalized free products J *g J which are flat 4-manifold groups with (5 = 0. The groups G2 *4, B 1, G2 *4, C 2 and G& Gq are all easily seen to be semidirect products of Gq with an infinite cyclic normal subgroup, on which Gq acts nontrivially. It follows easily th a t these three groups are in fact isomorphic, and so there is just one flat 4-manifold group with finite abelianization and holonomy isomorphic to (Z /2Z )2.

The above presentations of G2 *<p B 2 and Gq G4 are in fact equivalent; the function sending s to y, t to yw_1 and z to uy2u ~ l determines an isomorphism between these groups. Thus there is just one flat 4-manifold group with finite abelianization and holonomy isomorphic to Dg.

The above presentations of G'6*<^G2 and Gq^^B^ are also equivalent; the function sending x to x t~ x, y to yt and u to x y ~ l t determines an isomorphism between these groups (with inverse sending x to u y ~ lx ~ 2, y to u x ~ l and t to x u y ~ l ). (This isomorphism and the one in the paragraph above were found by Derek Holt, using the program described in [HR]). However the 2-quotients at class 3 of these groups and of Gq t ^ B ^ are distinct. (This was confirmed by Eamonn O ’Brien, who found th a t these quotients each have order 211, but have automorphism groups

FLAT 4-MANIFOLD GROUPS 39

of orders 220 and 221). Thus there are two flat 4-manifold groups with finite abelianization and holonomy isomorphic to (Z /2Z )3.

In summary, we have shown th a t there are 27 orientable flat 4-manifold groups (all with (3 > 0), 43 nonorientable flat 4-manifold groups with (3 > 0 and 4 (non- orientable) flat 4-manifold groups with (3 — 0.

7. Som e R e m a rk s o n F la t 4 -M an ifo ld s W h ic h a re S e ife rt F ib re d o r a re C o m p lex S u rfaces

It is not hard to see tha t most of the above groups admit normal subgroups of Hirsch length 2. The only exceptions are the semidirect products Gq xg Z where 6 = j , cej and abcej. In the orientable cases the corresponding 4-manifolds are then Seifert fibred (with general fibre a torus) over euclidean 2-orbifolds. (One of the two exceptions among orientable manifolds seems to have been overlooked in [Ue]).

If X is a complex algebraic surface which is homeomorphic to a flat 4-manifold then X is orientable and (3i(X) is even, so t t \{X) must be one of the eight flat 4-manifold groups 7F of orientable type and with ir = Z 4 or / ( 7r) = Z 2. In each case the holonomy group is cyclic, and so is conjugate (in G L +(4, R )) to a subgroup of G L(2, C). Thus all of these groups may be realized by complex analytic surfaces.It follows from the Enriques-Kodaira classification of surfaces th a t such surfaces are either complex tori or hyperelliptic, and hence th a t these groups are realized by algebraic surfaces. Moreover there are no complex analytic surfaces with (3\(X) odd which are homeomorphic to flat 4-manifolds. (See Chapter VI of [BPV]).

R e fe ren ces

[AJ] L. Auslander and F.E.A. Johnson, On a conjecture of C.T.C. Wall, J. London M ath. Soc. 14 (1976), 331-332.

[BPV] W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces,Ergebnisse der M athematik und ihrer Grenzgebiete 3 Folge Bd 4, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984.

[Bi] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Lecture Notes, London, 1976.

[B] H. Brown, R. Biilow, J. Neubiiser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space, J. Wiley, New York, 1978.

[Ca] E. Calabi, Closed, locally euclidean, 4-dimensional manifolds, Bull. Amer. Math. Soc. 63 (1957), 135.

[CS] L.S. Charlap and C.H. Sah, Compact flat riemannian 4-manifolds, pre-prelimin- ary report (1969).

[HI] J.A. Hillman, 2-Knots and their Groups, Australian M athematical Society Lecture Series 5, Cambridge University Press, 1989.

[H2] J.A. Hillman, The Algebraic Characterization of Geometric 4-Manifolds, London M athematical Society Lecture Note Series 198, Cambridge University Press, Cambridge, New York, Melbourne, 1994.

4 0 JONATHAN A. HILLMAN

[HR] D.F. Holt and S. Rees, Testing for isomorphism between finitely presented groups, in Groups, Combinatorics and Geometry (M.W. Liebeck and J. Saxl, eds.), Cambridge University Press, Cambridge, New York, Melbourne, 1992, pp. 459-475.

[Le] R. Levine, The Compact Euclidean Space Forms of Dimension 4, Doctoral dissertation, University of California, Berkeley, 1970.

[Ue] M. Ue, Geometric 4-manifolds in the sense of Thurston and Seifert 4-manifolds,I. J. Math. Soc. Japan 42 (1990), 511-540.

[Wo] J.A. Wolf, Spaces of Constant Curvature, fifth edition, Publish or Perish, Inc., Wilmington, Delaware, 1984.

[Zg] H. Zieschang, Finite Groups of Mapping Classes of Surfaces, Lecture Notes in M athematics 875, Springer-Verlag, Berlin, Heidelberg, New York, 1981.

[Zn] B. Zimmermann, On the Hantzsche-Wendt manifold, Mh. Math. 110 (1990), 321-327.

Jonathan A. HillmanDepartment of M athem atics and StatisticsThe University of SydneySydney NSW 2006AUSTRALIAhillm an-j@ m aths.su.oz.au