Groups: Topological Combinatorial and Arithmetic Aspects

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 We