Topological, Combinatorial and Arithmetic Aspects Edited by TW Müller

488 Pages·2012·3.32 MB·English

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. Mu¨ller
ii
Preface
iii
List of Authors and Participants
H. Abels, Fakulta¨t fu¨r Mathematik, Universita¨t Bielefeld, POB 100131, D-33501 Biele-
feld, Germany ([e-mail])
P. Abramenko, Department of Mathematics, University of Virginia, POB 400137 (Ker-
chof Hall), Charlottesville, VA 22904, USA (email: [e-mail])
S. I. Adian, Steklov Mathematical Institute, 42 ul. Vavilova, 117966 Moscow GSP-1,
Russia ([e-mail])
H. Behr, Fachbereich Mathematik, J. W. Goethe-Universita¨t, POB 111932, 60054
Frankfurt a. M., Germany ([e-mail])
R. Bieri, Fachbereich Mathematik, J. W. Goethe-Universita¨t, POB 111932, 60054
Frankfurt a. M., Germany ([e-mail])
M. Bridson, Department of Mathematics, Imperial College, 180 Queen’s Gate, London
SW72BZ ([e-mail])
K.-U. Bux, Department of Mathematics, Cornell University, 310 Malott Hall, Ithaka,
NY 14853-4201, USA (bux [e-mail])
P. J. Cameron, School of Mathematical Sciences, Queen Mary, University of London,
Mile End Road, London E14NS, UK ([e-mail])
I. M. Chiswell, School of Mathematical Sciences, Queen Mary, University of London,
Mile End Road, London E14NS, UK ([e-mail])
D. J. Collins, School of Mathematical Sciences, Queen Mary, University of London,
Mile End Road, London E14NS, UK ([e-mail])
A. Dress. Fakulta¨t fu¨r Mathematik, Universita¨t Bielefeld, POB 100131, D-33501 Biele-
feld, Germany ([e-mail])
R. Geoghegan, Department of Mathematical Sciences, SUNY, Binghamton, NY 13901,
USA ([e-mail])
R. I. Grigorchuk, Steklov Mathematical Institute, Gubkina Street 8, Moscow 117966,
Russia ([e-mail])
F.Grunewald,MathematischesInstitut,Heinrich-HeineUniversita¨t,D-40225Du¨sseldorf,
Germany ([e-mail])
H.Helling,Fakulta¨tfu¨rMathematik,Universita¨tBielefeld,POB100131,D-33501Biele-
feld, Germany ([e-mail])
W.Imrich,InstituteofAppliedMathematics,Montanuniversita¨tLeoben,A-8700Leoben,
Austria ([e-mail])
R. Kaplinsky, Jerusalem ORT College, Givat Ram, PB 39161, Jerusalem 91390, Israel
([e-mail])
I. Lysionok, Steklov Mathematical Institute, 42 ul. Vavilova, 117966 Moscow GSP-1,
Russia ([e-mail])
iv
A. Mann, Institute of Mathematics, The Hebrew University, Givat Ram, Jerusalem
91904, Israel ([e-mail])
J. Mennicke, Fakulta¨t fu¨r Mathematik, Universita¨t Bielefeld, POB 100131, D-33501
Bielefeld, Germany ([e-mail])
T. W. Mu¨ller, School of Mathematical Sciences, Queen Mary, University of London,
Mile End Road, London E14NS, UK ([e-mail])
V. Nekrashevych, Faculty of Mechanics and Mathematics, Kyiv Taras Shevchenko Uni-
versity, vul. Volodymyrska, 60, Kyiv, 01033, Ukraine ([e-mail])
J. R. Parker, Department of Mathematical Sciences, University of Durham, Durham
DH13LE, UK ([e-mail])
L. Reeves???????????
U. Rehmann, Fakulta¨t fu¨r Mathematik, Universita¨t Bielefeld, POB 100131, D-33501
Bielefeld, Germany ([e-mail])
B. Remy, Institut Fourier – UMR 5582, Universite Grenoble 1 – Joseph Fourier, 100 rue
desmaths,BP74–38402Saint-Martind’Heres,France([e-mail])
D. Segal, All Souls College, Oxford OX14AL, UK ([e-mail])
C. M. Series, Mathematics Institute, University of Warwick, Coventry, CV47AL, UK
(????????)
S. N. Sidki, Departamento de Matema´tica, Universidade de Bras´ılia, Bras´ılia -Df,
70.910-900, Brazil ([e-mail])
E. B. Vinberg, Department of Mechanics and Mathematics, Moscow State University,
Leninskie gory, 119899 Moscow, Russia ([e-mail])
J. S. Wilson, School of Mathematics and Statistics, University of Birmingham, Edgbas-
ton, Birmingham, B15 2TT, UK ([e-mail])
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. Mu¨ller
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 Infinite 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-
metricsonareductivegroupGoveralocalfieldaregiven. Theyareasfollows. Thefirst
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 of G by isometries on a metric space. That
these two pseudo-metrics on a group G are quasi-isometric holds in great generality.
The third pseudo-metric is defined using the operator norm for a representation ρ of
G. This pseudo-metric depends very much on the representation. But for a reductive
group over a local field it does not up to quasi-isometry. The fourth pseudo-metric is
given on a split torus over a local field K by valuations of the K –factors. The main
∗
result is that these four pseudo-metrics on a reductive group over a local field coincide
up to quasi-isometry. We thus have four different 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 field and an S–arithmetic subgroup Γ of G.
Then it was one of the main results of reduction theory to describe a fundamental
domain R for Γ 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 Γ G is a coarse isometry. He asked this question only for
→ \
the special case G = SL(n,R), Γ = SL(n,Z) and d the pseudo-metric on G coming
fromthestandardRiemannianmetriconthesymmetricspaceofG, thespaceofpositive
definiterealsymmetricn n–matrices. Wenowhaveapositiveanswerinfullgenerality,
×
for arbitrary reductive groups G over local fields, S–arithmetic subgroups Γ 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
Sections 5 and 6. This raises of course the question which pseudo-metrics are norm-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 pseudo-metrics
on reductive groups are norm-like. It is an open question whether the first 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
this is the case if G is a torus or if the rank r of a maximal split torus in the semi-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 first 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
1
2 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 Γ = SL(n,Z). Let T
be the subgroup of SL(n,R) of diagonal matrices t = diag(t1,...,tn) of determinant
one, a maximal R–split torus. The negative Weyl chamber is by definition the subset
C− = diag(t1,...,tn) T 0 < t1 t2 tn . A Siegel set R in SL(n,R) is, by
{ ∈ | ≤ ≤ ··· ≤ }
definition, 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
and L the Siegel set R is a set of representatives for G/Γ. So the natural map G G/Γ
→
restricts to a surjection π : R G/Γ. It has other nice properties, e.g., π is a proper
R
→ |
map. The question of Siegel mentioned above asked about the metric properties of π.
Let d be a right invariant pseudo-metric on G. Define a pseudo-metric d on G/Γ in the
natural way, i.e., d(g Γ,h Γ) = inf d(gγ,h) γ Γ . Now Siegel’s question was: is
{ | ∈ }
π : R G/Γ a coarse isometry? In other words: is there a constant C such that
→
d(g Γ,h Γ) d(g,h) d(g Γ,h Γ)+C
≤ ≤
for every pair g,h of points of R? Siegel himself showed in [S, Section 10] that this is
the case if we fix one variable, i.e., for every g G there is a constant C = C(g) such
∈
that the right inequality holds for every h R. It suffices to show this for one point
∈
g G.
∈
Here are the main steps of our proof that the answer is yes. We may assume that g and
h are in the negative Weyl chamber C and that d = dρ is the metric coming from a
− op
rational representation, see Section 5. We prove that, for γ Γ,
∈
d(gγ,h) d(a(gγ),h) d(w 1g,h) d(g,h)
−
→(I)≥ (→II)≥ (→III)≥
up to constants, where G = K A N, g = k(g) a(g) n(g), is the Iwasawa decomposition
· · · ·
and γ BwB in the Bruhat decomposition with w an element of the Weyl group S .
n 1
∈ −
Note that (III) is a very special property of reflection groups. It does for example not
hold for g,h in the fundamental domain of a finite rotation group and w in this group.
An important step in the proof of (II) is
(II) a(gγ) = w 1gw+r
(cid:48) −
where r is up to a compact error term the exponential of a positive linear combination
of Σ where Σ = α Φ+ w 1αw Φ+ and Φ+ is the set of positive roots.
w−1 w−1 −
{ ∈ | ∈ }
That we found (II) is due to discussions with Alex Eskin who showed us a geometric
(cid:48)
picture of this fact.
Let us point out the following features of this proof. It is different from both Ding’s
[2] which is by induction on n, and from Leuzinger’s [L] which uses Tits buildings and
facts about the geometry of symmetric and locally symmetric spaces, in particular their
geometry at infinity. Our proof works in full generality, for arbitrary local fields and
arbitrary S–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 first need to recall some concepts concerning metric spaces.
2.1. Let X be a set. A function d : X X R is called a pseudo-metric (on X) if d is
× →
non-negative, zero on the diagonal, symmetric and fulfills 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). So
d
B (x,r) = y X : d(x,y) r .
d
∈ ≤
(cid:169) (cid:170)
2.2. Let (X,d) and (X ,d) be pseudo-metric spaces. A map f : X X is called a
(cid:48) (cid:48) (cid:48)
→
quasi-isometry if there are real numbers C > 0 and C such that
1 2
C 1 d(x,y) C d(f(x),f(y)) C d(x,y)+C
1− · − 2 ≤ (cid:48) ≤ 1 · 2
and X = B (f(x),C ). Thus, for every point x X there is a point x X such
(cid:48) x X d(cid:48) 2 (cid:48) ∈ (cid:48) ∈
that d(x,f(x∈)) C . Define a map g : X X by choosing for every x X a point
(cid:48) 2 (cid:48) (cid:48)
(cid:83) ≤ → ∈
x = g(x) with this property. Then g : X X is a quasi-isometry, actually with the
(cid:48) (cid:48)
→
same multiplicative constant C , and we have d(x,gf(x)) C and d(x,fg(x)) C
1 2 (cid:48) (cid:48) (cid:48) 2
≤ ≤
for every x X and x X .
(cid:48) (cid:48)
∈ ∈
2.3. A map f : X X between pseudo-metric spaces (X,d) and (X ,d) is called
(cid:48) (cid:48) (cid:48)
→
a coarse isometry if f is a quasi-isometry and the multiplicative constant C can be
1
chosentoequal1. Equivalently, thefunction(x,y) d (f(x),f(y)) d(x,y)isbounded
(cid:48)
(cid:55)→ −
on X X and every point of X is at bounded distance from f(X). Finally, f :
(cid:48)
×
X X is called an isometry if both these bounds are zero, i.e., if f is surjective and
(cid:48)
→
d(f(x),f(y)) = d(x,y) for every x,y in X. If f is a (coarse) isometry, then so is any
(cid:48)
map g : X X considered above. It follows that if there is a (quasi-, coarse) isometry
(cid:48)
→
from X to X then there is one from X to X. Two pseudo-metrics on the same set are
(cid:48) (cid:48)
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. Define 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, zero
4 H. Abels
at the identity element, symmetric and fulfills the triangle inequality, i.e.,
f(g) 0 for every g G,
≥ ∈
f(e) = 0 for the identity element e,
f(g) = f(g 1) for every g G and
−
∈
f(gh) f(g)+f(h) for every g,h in G.
≤
Conversely, given a function f with these properties then d(g,h) := f(g 1h), resp.
−
d(g,h) = f(hg 1), defines the unique left (right) invariant pseudo-metric d on G such
−
that d(e,g) = f(g) for every g 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 Σ be a set of generators of G. Then the word length (cid:96) (g) of
Σ
an element g G with respect to Σ is defined as
∈
(cid:96) (g) = inf r : g = aε1...aεr, a Σ, ε +1, 1 .
Σ 1 r i ∈ i ∈ { − }
The function (cid:96) has the p(cid:169)roperties stated above and furthermore (cid:96)(cid:170)(g) = 0 implies
Σ Σ
g = e. So d (g,h) := (cid:96) (g 1h) defines a left invariant metric d on G, which is called
Σ Σ − Σ
the word metric associated with Σ. The ball of radius r with center e is
B (e,r) = (Σ Σ 1)r = aε1...aεr : a Σ , ε +1, 1 ,
dΣ ∪ − 1 r i ∈ i ∈ { − }
and thus consists of all words of lengt(cid:169)h at most r with respect to the alpha(cid:170)bet Σ Σ 1.
−
∪
The word metric d depends of course on Σ. But if Σ and Σ are both finite sets of
Σ (cid:48)
generators of G then d and d are quasi-isometric, since if (cid:96) (Σ) is bounded by C
Σ Σ(cid:48) Σ (cid:48) 1
then d C d . Similarly:
Σ 1 Σ(cid:48)
≤ ·
3.1. Lemma. Let G be a locally compact topological group and let Σ and Σ be compact
(cid:48)
sets of generators of G. Then the word metrics d and d on G are quasi-isometric.
Σ Σ(cid:48)
They are actually Lipschitz equivalent, i.e., the additive constant C in the definition of
2
quasi-isometry may be chosen equal to zero.
By the preceding argument it suffices to show the following.
3.2. Lemma. Let G be a locally compact topological group and let Σ be a compact set
of generators of G. Then every compact subset of G has bounded word length (cid:96) .
Σ
Proof. The sequence of compact subsets A = B (e,n) = (Σ Σ 1)n of G covers the
n dΣ ∪ −
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 identity
n 2n
⊂
element e, since A contains U U 1. If now K is a compact subset of G there is a
2n −
·
finite subset M of K such that M A contains K. Thus (cid:96) (K) (cid:96) (M)+2n. (cid:164)
2n Σ Σ
· ≤
Reductive Groups as Metric Spaces 5
3.3. Remark. BothLemmas3.1and3.2remaintrueifΣandΣ arerelativelycompact
(cid:48)
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
additive group R. The word length (cid:96)Σ(cid:48) corresponding to the set of generators Σ(cid:48) = [0,1]
is (cid:96) (x) = x , the smallest integer x . Consider the following set of generators
Σ(cid:48)
(cid:100)| |(cid:101) ≥ | |
Σ. There is a basis B of the Q–vector space R such that B [0,1] and B contains
⊂
for every n ∈ N an element bn with 0 ≤ bn ≤ n1. Such a basis can be obtained from a
given basis of R over Q by multiplying every basis element with an appropriate rational
number. Put Σ = q b : b B, q Q [0,1] [0,1]. Then Σ is a set of generators
{ · ∈ ∈ ∩ } ⊂
of R, contained in [0,1] but (cid:96)Σ is unbounded on [0,1], since (cid:96)Σ(nbn) = n. In fact, for
every real number x = b Bqb ·b with qb ∈ Q, we have (cid:96)Σ(x) = b B(cid:100)|qb|(cid:101).
∈ ∈
Here is a geometric app(cid:80)roach to the word metric. (cid:80)
3.4. Definition. A pseudo-metric d on a set X is called a coarse path pseudo-metric if
there is a real number C such that for every pair of points x,y in X there is a sequence
x = x ,x ,...,x = y for which d(x ,x ) C for i = 1,...,t and
0 1 t i 1 i
− ≤
t
d(x,y) d(x ,x ) C.
i 1 i
≥ − −
i=1
(cid:88)
In other words, the triangle inequality d(x,y) t d(x ,x ) is in fact an equality
≤ i=1 i−1 i
up to a bounded error.
(cid:80)
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 g G there is a sequence g ,...,g of elements of G
1 t
such that g = g g , f(g ) C∈for i = 1,...,t and f(g) t f(g ) C. The
1 ····· t i ≤ ≥ i=1 i −
equivalenceisseenasfollows. Startingwithg Gtakeasequencex = e,x ,...,x = g
0 1 t
as above and put g = x 1 x . Conversely, f∈or x,y in G take a s(cid:80)equence g ,...,g as
i −i 1 · i 1 t
above for g = x 1y and p−ut x = x g g .
− i 1 i
· ·····
3.6. Example. Awordmetricd onagroupisacoarsepathmetric, sincebydefinition
Σ
C = 1, B(e,1) = Σ Σ 1 e 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 Σ of G and
a bounded function ω : Σ [0, ) on Σ we can define a weighted word length on G by
→ ∞
t
(cid:96)Σ,ω(g) = inf ω(gi) : t ∈ N∪{0}, g = g1ε1 ·····gtεt, gi ∈ Σ, εi ∈ {+1,−1} .
(cid:189) i=1 (cid:190)
(cid:88)
Then (cid:96) has all the properties of 2.4 so that d (g,h) := (cid:96) (g 1h) defines a left
Σ,ω Σ,ω Σ,ω −
invariant pseudo-metric on G 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-
Σ,ω
erty that d(e,g) ω(g) for g Σ.
≤ ∈
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.