Groups: Topological, Combinatorial

and Arithmetic Aspects

Proceedings of a conference, held 15 { 21 August 1999 at the

University of Bielefeld.

Supported by the Deutsche Forschungsgemeinschaft (DFG)

through Sonderforschungsbereich 343, University of Bielefeld.

Edited by T. W. Muller˜ii

Prefaceiii

List of Authors and Participants

H. Abels, Fakult˜at fur˜ Mathematik, Universit˜at Bielefeld, POB 100131, D-33501 Biele-

feld, Germany (abels@mathematik.uni-bielefeld.de)

P. Abramenko, Department of Mathematics, University of Virginia, POB 400137 (Ker-

chof Hall), Charlottesville, VA 22904, USA (email: pa8e@virginia.edu)

S. I. Adian, Steklov Mathematical Institute, 42 ul. Vavilova, 117966 Moscow GSP-1,

Russia (adian@log.mian.su)

H. Behr, Fachbereich Mathematik, J. W. Goethe-Universit˜at, POB 111932, 60054

Frankfurt a. M., Germany (helmut.behr@math.uni-frankfurt.de)

R. Bieri, Fachbereich Mathematik, J. W. Goethe-Universit˜at, POB 111932, 60054

Frankfurt a. M., Germany (bieri@math.uni-frankfurt.de)

M. Bridson, Department of Mathematics, Imperial College, 180 Queen’s Gate, London

SW72BZ (m.bridson@ic.ac.uk)

K.-U. Bux, Department of Mathematics, Cornell University, 310 Malott Hall, Ithaka,

NY 14853-4201, USA (bux 2002@kubux.net)

P. J. Cameron, School of Mathematical Sciences, Queen Mary, University of London,

Mile End Road, London E14NS, UK (p.j.cameron@qmul.ac.uk)

I. M. Chiswell, School of Sciences, Queen Mary, University of London,

Mile End Road, London E14NS, UK (i.m.chiswell@qmul.ac.uk)

D. J. Collins, School of Mathematical Sciences, Queen Mary, University of London,

Mile End Road, London E14NS, UK (d.j.collins@qmul.ac.uk)

A. Dress. Fakult˜at fur˜ Mathematik, Universit˜at Bielefeld, POB 100131, D-33501 Biele-

feld, Germany (dress@mathematik.uni-bielefeld.de)

R. Geoghegan, Department of Mathematical Sciences, SUNY, Binghamton, NY 13901,

USA (ross@math.binghamton.edu)

R. I. Grigorchuk, Steklov Mathematical Institute, Gubkina Street 8, Moscow 117966,

Russia (grigorch@mi.ras.ru)

F.Grunewald,MathematischesInstitut,Heinrich-HeineUniversit˜at,D-40225Dusseldorf,˜

Germany (fritz@math.uni-duesseldorf.de)

H.Helling,Fakult˜atfur˜ Mathematik,Universit˜atBielefeld,POB100131,D-33501Biele-

feld, Germany (helling@mathematik.uni-bielefeld.de)

W.Imrich,InstituteofAppliedMathematics,Montanuniversit˜atLeoben,A-8700Leoben,

Austria (imrich@unileoben.ac.at)

R. Kaplinsky, Jerusalem ORT College, Givat Ram, PB 39161, Jerusalem 91390, Israel

(rkaplins@mail.ort.org.il)

I. Lysionok, Steklov Mathematical Institute, 42 ul. Vavilova, 117966 Moscow GSP-1,

Russia (lysionok@euclid.mi.ras.ru)iv

A. Mann, Institute of Mathematics, The Hebrew University, Givat Ram, Jerusalem

91904, Israel (mann@vms.huji.ac.il)

J. Mennicke, Fakult˜at fur˜ Mathematik, Universit˜at Bielefeld, POB 100131, D-33501

Bielefeld, Germany (mennicke@mathematik.uni-bielefeld.de)

T. W. Muller,˜ School of Mathematical Sciences, Queen Mary, University of London,

Mile End Road, London E14NS, UK (t.w.muller@qmul.ac.uk)

V. Nekrashevych, Faculty of Mechanics and Mathematics, Kyiv Taras Shevchenko Uni-

versity, vul. Volodymyrska, 60, Kyiv, 01033, Ukraine (nazaruk@ukrpack.net)

J. R. Parker, Department of Mathematical Sciences, University of Durham, Durham

DH13LE, UK (j.r.parker@durham.ac.uk)

L. Reeves???????????

U. Rehmann, Fakult˜at fur˜ Mathematik, Universit˜at Bielefeld, POB 100131, D-33501

Bielefeld, Germany (rehmann@mathematik.uni-bielefeld.de)

B.Remy, InstitutFourier{UMR5582, UniversiteGrenoble1{JosephFourier, 100rue

desmaths,BP74{38402Saint-Martind’Heres,France(bertrand.remy@ujf-grenoble.fr)

D. Segal, All Souls College, Oxford OX14AL, UK (dan.segal@all-souls.oxford.ac.uk)

C. M. Series, Mathematics Institute, University of Warwick, Coventry, CV47AL, UK

(????????)

S. N. Sidki, Departamento de Matem¶atica, Universidade de Bras¶‡lia, Bras¶‡lia -Df,

70.910-900, Brazil (sidki@mat.unb.br)

E. B. Vinberg, Department of Mechanics and Mathematics, Moscow State University,

Leninskie gory, 119899 Moscow, Russia (vinberg@ebv.pvt.msu.su)

J. S. Wilson, School of Mathematics and Statistics, University of Birmingham, Edgbas-

ton, Birmingham, B15 2TT, UK (jsw@for.mat.bham.ac.uk)v

Contents

H. Abels

Reductive Groups as Metric Spaces 1

P. Abramenko

Finiteness Properties of Groups Acting on Twin Buildings ?

H. Behr

S-Arithmetic Groups in the Function Field Case I ?

R. Bieri and R. Geoghegan

Controlled Topology and Group Actions ?

K.-U. Bux

Finiteness Properties of Soluble S-Arithmetic Groups { A Survey ?

P. J. Cameron

Topology in Permutation Groups ?

I. M. Chiswell

Euler Characteristics of Discrete Groups ?

D. J. Collins

Intersection of Magnus Subgroups of One-Relator Groups ?

R. I. Grigorchuk and J. S. Wilson

A Minimality Property of Certain Branch Groups ?

H. Helling

Lattices with Non-Integral Character ?

A. Mann

Some Applications of Probability in Group Theory ?

T. W. Mul˜ ler

Parity Patterns in Hecke Groups and Fermat Primes ?

V. Nekrashevych and S. Sidki

Automorphisms of the Binary Tree: State-Closed Subgroups and

Dynamics of 1=2-Endomorphisms ?

J. R. Parker and C. Series

The Mapping Class Group of the Twice Punctured Torus ?

B. Remy

Kac-Moody Groups: Split and Relative Theories. Lattices ?

D. Segal

On the Images of Inﬂnite Groups ?

E. B. Vinberg and R. Kaplinsky

Pseudo-Finite Generalized Triangle Groups ?Reductive Groups as Metric Spaces

by

H. Abels

1. Introduction

In this paper four descriptions of one and the same quasi-isometry class of pseudo-

metricsonareductivegroupGoveralocalﬂeldaregiven. Theyareasfollows. Theﬂrst

one is the word metric corresponding to a compact set of generators of G. The second

one is the pseudo-metric given by the action ofG by isometries on a metric space. That

these two pseudo-metrics on a group G are quasi-isometric holds in great generality.

The third is deﬂned using the operator norm for a representation ‰ of

G. This pseudo-metric depends very much on the representation. But for a reductive

group over a local ﬂeld it does not up to quasi-isometry. The fourth pseudo-metric is

⁄given on a split torus over a local ﬂeld K by valuations of the K {factors. The main

result is that these four pseudo-metrics on a reductive group over a local ﬂeld coincide

up to quasi-isometry. We thus have four diﬁerent descriptions of one and the same very

natural and distinguished quasi-isometry class of pseudo-metrics.

ThispapercontainsfoundationalmaterialforjointworkinprogresswithG.A.Margulis

on the following two topics. One is work on the following question of C. L. Siegel’s.

Given a reductive group G over a local ﬂeld and an S{arithmetic subgroup ¡ of G.

Then it was one of the main results of reduction theory to describe a fundamental

domain R for ¡ in G, a so called Siegel domain. Siegel asked in his Japan lectures [S,

end of Section 10] on reduction theory of 1959, if { in our terminology, see Section 2.3

{ the natural map R ! ¡nG is a coarse isometry. He asked this question only for

the special case G = SL(n;R), ¡ = SL(n;Z) and d the pseudo-metric on G coming

fromthestandardRiemannianmetriconthesymmetricspaceofG,thespaceofpositive

deﬂniterealsymmetricn£n{matrices. Wenowhaveapositiveanswerinfullgenerality,

for arbitrary reductive groups G over local ﬂelds, S{arithmetic subgroups ¡ and for

pseudo-metrics d on G which are norm-like. We call a pseudo-metric on G norm{like

if it is coarsely isometric to a metric coming from the operator norm of a rational

representation, or, equivalently, coming from a norm on a maximal split torus, see

Sections5and6. Thisraisesofcoursethequestionwhichpseudo-metricsarenorm-like.

Note that coarse isometry is a much stricter equivalence relation among pseudo-metrics

than quasi-isometry. We show in this paper that the three last types of

on reductive groups are norm-like. It is an open question whether the ﬂrst one, namely

the word metric, or, more generally (Section 3.8), any coarse path pseudo-metric, gives

a norm-like pseudo-metric. In joint work in progress with G. A. Margulis we show that

thisisthecaseifGisatorusoriftherankr ofamaximalsplittorusinthesemi-simple

part of G is equal to one. This is probably even true for r = 2:

The question of Siegel has an interesting history. A ﬂrst positive answer was given

by Borel in [1]. It was discovered much later [JM] that the proof contains a gap. It

occurs on pp. 550 { 552, (12) does not imply (14), but (14) is essential to prove (5),

the main inequality. There are now proofs for Siegel’s conjecture, in its original form

12 H. Abels

[2] and more generally for real reductive groups G; ordinary arithmetic subgroups and

the pseudo-metric d coming from the symmetric space [4, 6].

Here are some more details about our approach to Siegel’s question. For the sake of

exposition we restrict ourselves to the case G = SL(n;R) and ¡ = SL(n;Z): Let T

be the subgroup of SL(n;R) of diagonal matrices t = diag(t ;:::;t ) of determinant1 n

one, a maximalR{split torus. The negative Weyl chamber is by deﬂnition the subset

¡C =fdiag(t ;:::;t )2T j 0<t •t •¢¢¢•t g: A Siegel set R in SL(n;R) is, by1 n 1 2 n

¡deﬂnition, a subset of G of the form K¢C ¢L; where K and L are compact subsets of

G: The main result of reduction theory for this case states that for appropriate sets K

andL the Siegel setR is a set of representatives forG=¡: So the natural mapG!G=¡

restricts to a surjection… :R!G=¡: It has other nice properties, e.g., …j is a properR

map. The question of Siegel mentioned above asked about the metric properties of …:

Let d be a right invariant pseudo-metric on G: Deﬂne a pseudo-metric d on G=¡ in the

natural way, i.e., d(g ¡;h ¡) = inffd(g?;h)j ?2 ¡g: Now Siegel’s question was: is

… :R!G=¡ a coarse isometry? In other words: is there a constant C such that

d(g ¡;h ¡)•d(g;h)•d(g ¡;h ¡)+C

for every pair g;h of points of R? Siegel himself showed in [S, Section 10] that this is

the case if we ﬂx one variable, i.e., for every g2 G there is a constant C = C(g) such

that the right inequality holds for every h2 R: It su–ces to show this for one point

g2G:

Here are the main steps of our proof that the answer is yes. We may assume thatg and

¡ ‰h are in the negative Weyl chamber C and that d = d is the metric coming from aop

rational representation, see Section 5. We prove that, for ?2 ¡;

¡1d(g?;h)!‚d(a(g?);h)!‚d(w g;h) !‚d(g;h)

(I) (II) (III)

uptoconstants, whereG =K¢A¢N,g =k(g)¢a(g)¢n(g);istheIwasawadecomposition

and?2BwB intheBruhatdecompositionwithw anelementoftheWeylgroupS :n¡1

Note that (III) is a very special property of re ection groups. It does for example not

hold for g;h in the fundamental domain of a ﬂnite rotation group and w in this group.

An important step in the proof of (II) is

0 ¡1(II) a(g?) =w gw+r

where r is up to a compact error term the exponential of a positive linear combination

+ ¡1 + +of § ¡1 where § ¡1 =fﬁ2 ' j w ﬁw2 ' g and ' is the set of positive roots.w w

0That we found (II) is due to discussions with Alex Eskin who showed us a geometric

picture of this fact.

Let us point out the following features of this proof. It is diﬁerent from both Ding’s

[2] which is by induction on n, and from Leuzinger’s [L] which uses Tits buildings and

factsaboutthegeometryofsymmetricandlocallysymmetricspaces, inparticulartheir

geometry at inﬂnity. Our proof works in full generality, for arbitrary local ﬂelds and

arbitraryS{arithmetic subgroups. Also we admit arbitrary norm-like metrics, not only

those coming from the symmetric space or the Bruhat{Tits building. Finally it gives

further information concerning reduction theory, namely the inequalities stated above.Reductive Groups as Metric Spaces 3

2. Metrics

We ﬂrst need to recall some concepts concerning metric spaces.

2.1. LetX be a set. A functiond :X£X!R is called a pseudo-metric (onX) ifd is

non-negative, zero on the diagonal, symmetric and fulﬂlls the triangle inequality, i.e., if

d(x;y) ‚ 0 for every x;y in X

d(x;x) = 0 for every x in X

d(x;y) = d(y;x) for every x;y in X

d(x;y)+d(y;z) ‚ d(x;z) for every x;y;z in X:

So a pseudo-metric on X is a metric on X if and only if d(x;y) = 0 implies x = y. A

pair (X;d) consisting of a set X and a (pseudo-) metric d on X is called a (pseudo-)

metric space. In a pseudo-metric space (X;d) the ball of radius r with center x is

denoted B (x;r) or B(x;r). Sod

' “

B (x;r) = y2X : d(x;y)•r :d

0 0 02.2. Let (X;d) and (X ;d) be pseudo-metric spaces. A map f : X ! X is called a

quasi-isometry if there are real numbers C > 0 and C such that1 2

¡1 0C ¢d(x;y)¡C •d(f(x);f(y))•C ¢d(x;y)+C2 1 21

S

0 0 0andX = B 0(f(x);C ). Thus,foreverypointx 2X thereisapointx2X suchd 2x2X

0 0 0that d(x;f(x))• C . Deﬂne a map g : X ! X by choosing for every x 2 X a point2

0 0x = g(x) with this property. Then g : X ! X is a quasi-isometry, actually with the

0 0 0same multiplicative constant C , and we have d(x;gf(x))•C and d(x;fg(x))•C1 2 2

0 0for every x2X and x 2X .

0 0 02.3. A map f : X ! X between pseudo-metric spaces (X;d) and (X ;d) is called

a coarse isometry if f is a quasi-isometry and the multiplicative constant C can be1

0chosentoequal1. Equivalently,thefunction(x;y)7!d(f(x);f(y))¡d(x;y)isbounded

0on X£ X and every point of X is at bounded distance from f(X). Finally, f :

0X! X is called an isometry if both these bounds are zero, i.e., if f is surjective and

0d(f(x);f(y)) = d(x;y) for every x;y in X. If f is a (coarse) isometry, then so is any

0mapg :X !X considered above. It follows that if there is a (quasi-, coarse) isometry

0 0from X to X then there is one from X toX. Two pseudo-metrics on the same set are

called (quasi-, coarsely) isometric if the identity map is a (quasi-, coarse) isometry. It

follows that these relations are equivalence relations between pseudo-metric spaces and

also between pseudo-metrics on the same set.

2.4. We will mainly be interested in pseudo-metrics on groups. So let G be a group.

A pseudo-metric d on G will be called left invariant (right invariant) if every left

translation (right translation) is an isometry. So d is left invariant on G if and only if

d(gh ;gh ) = d(h ;h ) for every g;h ;h in G. Deﬂne a function f on G by f(g) =1 2 1 2 1 2

d(e;g). If d is a left (right) invariant pseudo-metric on G, then f is non-negative, zero4 H. Abels

at the identity element, symmetric and fulﬂlls the triangle inequality, i.e.,

f(g) ‚ 0 for every g2G;

f(e) = 0 for the identity element e;

¡1f(g) = f(g ) for every g2G and

f(gh) • f(g)+f(h) for every g;h in G:

¡1Conversely, given a function f with these properties then d(g;h) := f(g h), resp.

¡1d(g;h) = f(hg ), deﬂnes the unique left (right) invariant pseudo-metric d on G such

that d(e;g) = f(g) for every g 2 G. A function f on G with these properties is

sometimes called a norm on G. But we want to reserve the term \norm" for a more

special situation.

3. The word metric

Let G be a group and let § be a set of generators of G. Then the word length ‘ (g) of§

an element g2G with respect to § is deﬂned as

' “

" "1 r‘ (g) = inf r : g =a :::a ; a 2 §; " 2f+1;¡1g :§ i i1 r

The function ‘ has the properties stated above and furthermore ‘ (g) = 0 implies§ §

¡1g = e. So d (g;h) := ‘ (g h) deﬂnes a left invariant metric d on G, which is called§ § §

the word metric associated with §. The ball of radius r with center e is

' “

¡1 r " "1 rB (e;r) = (§[§ ) = a :::a : a 2 § ; " 2f+1;¡1g ;d i i§ 1 r

¡1and thus consists of all words of length at mostr with respect to the alphabet §[§ .

0The word metric d depends of course on §. But if § and § are both ﬂnite sets of§

0

0generators of G then d and d are quasi-isometric, since if ‘ (§) is bounded by C§ § § 1

0then d •C ¢d . Similarly:§ 1 §

03.1. Lemma. LetG be a locally compact topological group and let § and § be compact

sets of generators of G. Then the word metrics d and d 0 on G are quasi-isometric.§ §

They are actually Lipschitz equivalent, i.e., the additive constant C in the deﬂnition of2

quasi-isometry may be chosen equal to zero.

By the preceding argument it su–ces to show the following.

3.2. Lemma. Let G be a locally compact topological group and let § be a compact set

of generators of G. Then every compact subset of G has bounded word length ‘ .§

¡1 nProof. The sequence of compact subsets A = B (e;n) = (§[§ ) of G covers then d§

locally compact space G. So one of them contains a non-empty open subset U of G by

the Baire category theorem, say U‰A . Then A is a neighbourhood of the identityn 2n

¡1element e, since A contains U¢U . If now K is a compact subset of G there is a2n

ﬂnite subset M of K such that M¢A contains K. Thus ‘ (K)•‘ (M)+2n. ⁄2n § §Reductive Groups as Metric Spaces 5

03.3. Remark. BothLemmas3.1and3.2remaintrueif§and§ arerelativelycompact

sets of generators of G which contain a non-empty open subset of G, as follows from

the second part of the proof of Lemma 3.2. But Lemma 3.2, and hence Lemma 3.1, is

not true for an arbitrary relatively compact set of generators of G; e.g., let G be the

0additive groupR. The word length‘ 0 corresponding to the set of generators § = [0;1]§

is ‘ 0(x) =djxje, the smallest integer‚jxj. Consider the following set of generators§

§. There is a basis B of the Q{vector space R such that B ‰ [0;1] and B contains

1for every n2N an element b with 0• b • . Such a basis can be obtained from an n n

given basis ofR overQ by multiplying every basis element with an appropriate rational

number. Put § =fq¢b : b2B; q2Q\[0;1]g‰ [0;1]. Then § is a set of generators

ofR, contained in [0;1] but ‘ is unbounded on [0;1], since ‘ (nb ) = n. In fact, for§ § nP P

every real number x = q ¢b with q 2Q, we have ‘ (x) = djqje.b b § bb2B b2B

Here is a geometric approach to the word metric.

3.4. Deﬂnition. A pseudo-metricd on a setX is called a coarse path pseudo-metric if

there is a real numberC such that for every pair of pointsx;y inX there is a sequence

x =x ;x ;:::;x =y for which d(x ;x )•C for i = 1;:::;t and0 1 t i¡1 i

tX

d(x;y)‚ d(x ;x )¡C:i¡1 i

i=1

P

tIn other words, the triangle inequality d(x;y)• d(x ;x ) is in fact an equalityi¡1 ii=1

up to a bounded error.

3.5. A left invariant pseudo-metric d on a group G is a coarse path pseudo-metric if

and only if the function f with f(g) = d(e;g) has the following property. There is a

real number C such that for every g2G there is a sequence g ;:::;g of elements of G1 tPt

such that g = g ¢¢¢¢¢g , f(g )• C for i = 1;:::;t and f(g)‚ f(g )¡C. The1 t i ii=1

equivalenceisseenasfollows. Startingwithg2Gtakeasequencex =e;x ;:::;x =g0 1 t

¡1as above and put g = x ¢x . Conversely, for x;y in G take a sequence g ;:::;g asi i 1 ti¡1

¡1above for g =x y and put x =x¢g ¢¢¢¢¢g .i 1 i

3.6. Example. Awordmetricd onagroupisacoarsepathmetric,sincebydeﬂnition§

¡1C = 1, B(e;1) = §[§ [feg and the error in the triangle inequality is zero with

notation as in 3.4.

3.7. One can generalize this example as follows. Given a set of generators § of G and

a bounded function ! : §! [0;1) on § we can deﬂne a weighted word length on G by

‰ ?tX

" "1 t‘ (g) = inf !(g ) : t2N[f0g; g =g ¢¢¢¢¢g ; g 2 §; " 2f+1;¡1g :§;! i i i1 t

i=1

¡1Then ‘ has all the properties of 2.4 so that d (g;h) := ‘ (g h) deﬂnes a left§;! §;! §;!

invariant pseudo-metric onG which is in fact a coarse path pseudo-metric, as is readily

seen. Furthermore, d is the supremum of the pseudo-metrics d on X with the prop-§;!

erty that d(e;g)•!(g) for g2 §.

3.8. The importance of this generalization lies in the following fact: every left invari-

ant coarse path pseudo-metric is a weighted word pseudo-metric up to coarse isometry.