Trees

149 Pages·1980·11.111 MB·English

by  Jean-Pierre Serre (auth.)

Jean -Pierre Serre
TREES
Translated from the French by John Stillwell
Springer-Verlag
Berlin Heidelberg New York 1980
Jean-Pierre Serre
College de France, Chaire d'Algebre et Geometrie
F-75231 Paris Cedex 05
Translator:
John Stillwell
Dept. of Mathematics, Monash University
AUS-Clayton, Victoria 3168
Title of the French Original Edition:
Arbres, Amalgames, SLz. Asterisque no. 46,
Soc. Math. France, 1977. ISSN 0303-1179
ISBN-13: 978-3-642-61858-1 e-ISBN-13: 978-3-642-61856-7
DOl: 10.1007/978-3-642-61856-7
Library of Congress Cataloging in Publication Data. Serre. Jean-Pierre. Trees. Translation
of Arbres, Amalgames, SL2. Bibliography: p. Includes index.!. Linear algebraic groups.
2. Free groups. 3. Trees (Graph theory) I. Title. QAI71.S52713. 512'2. 80-16116
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@ by Springer-Verlag Berlin Heidelberg 1980
Softcover reprint of the hardcover 1st edition 1980
Typesetting: Buchdruckerei Dipl.-lng. Schwarz' Erben KG, Zwettl. Printing and binding:
Konrad Triltsch, Wiirzburg
2141/3140-5432 I 0
Table of Contents
Introduction. . . . . . . . . . . . VII
Chapter I. Trees and Amalgams.
§ 1 Amalgams. . . . . . . . . . 1
1.1 Direct limits. . . . . . . 1
1.2 Structure of amalgams. 2
1.3 Consequences of the structure theorem. 5
1.4 Constructions using amalgams 8
1.5 Examples 11
§2 Trees. . . . 13
2.1 Graphs . 13
2.2 Trees . . 17
2.3 Subtrees of a graph 21
§3 Trees and free groups. . 25
3.1 Trees of representatives 25
3.2 Graph of a free group . 26
3.3 Free actions on a tree . 27
3.4 Application: Schreier's theorem. 29
Appendix: Presentation of a group of homeomorphisms 30
§4 Trees and amalgams . . . . . . . . . . . . . . . . 32
4.1 The case of two factors . . . . . . . . . . . . 32
4.2 Examples of trees associated with amalgams. 35
4.3 Applications . . . . . . . . . . . . . . . . . . 36
4.4 Limit of a tree of groups . . . . . . . . . . . 37
4.5 Amalgams and fundamental domains (general case) 38
§5 Structure of a group acting on a tree . . . . . 41
5.1 Fundamental group of a graph of groups. . . . 41
5.2 Reduced words. . . . . . . . . . . . . . . . . . . 45
5.3 Universal covering relative to a graph of groups 50
5.4 Structure theorem . . . . . . . . 54
5.5 Application: Kurosh's theorem. . . . . . . . . . 56
§6 Amalgams and fixed points. . . . . . . . . . . . . . . 58
6.1 The fixed point property for groups acting on trees 58
6.2 Consequences of property (FA). . . . . . . . . . .. 59
VI Table of Contents
6.3 Examples . . . . . . . . . . . . . . . . . . . 60
6.4 Fixed points of an automorphism of a tree 61
6.5 Groups with fixed points (auxiliary results) 64
6.6 The case of SL3(Z), 67
Chapter II. SL2. . . . . . . 69
§ 1 The tree of SL over a local field 69
2
1.1 The tree. . . . . . . . . . . . 69
1.2 The groups GL(V) and SL(V) 74
1.3 Action of GL(V) on the tree of V; stabilizers. 76
1.4 Amalgams. . . . . 78
1.5 Ihara's theorem. . . . . . . . . 82
1.6 Nagao's theorem. . . . . . . . 85
1.7 Connection with Tits systems . 89
§2 Arithmetic subgroups of the groups GL and SL over a
2 2
function field of one variable . . . . . . . . . . . . . . .. 96
2.1 Interpretation of the vertices of nX as classes of
vector bundles of rank 2 over C . . . . . . . . 96
2.2 Bundles of rank 1 and decomposable bundles. 99
2.3 Structure of nX . 103
2.4 Examples . . . . 111
r . .
2.5 Structure of 117
2.6 Auxiliary results . 120
r:
2.7 Structure of case of a finite field 124
2.8 Homology. . . . . . . . . . . 125
2.9 Euler-Poincare characteristic 131
Bibliography. 137
Index . . . . 141
Introduction
The starting point of this work has been the theorem ofIhara [16],
according to which every torsion-free subgroup G of SLz(Qp) is a
free group. This striking result was at the time (1966) the only one
known concerning the structure of discrete subgroups of p-adic
groups.
Ihara's proof is combinatorial; it uses, in a somewhat mys
terious way, a decomposition of SLz(Qp) as an amalgam of two
copies of SLz(Zp). But topology suggests a natural way to prove
that a group G is free: it suffices to make G act freely ("without fixed
points") on a tree X; the group G may then be identified with the
fundamental group nl(G\X) of the quotient graph G\X, a group
which is obviously free. Interpreted from this point of view, Ihara's
proof amounts to taking for X the tree associated with the amalgam
mentioned above; this tree, the tree of SLz over the field Qp, then
appears as a very special case of a Bruhat-Tits building ([37], [38]),
the p-adic analogue of the symmetric homogeneous spaces of real
Lie groups.
One is therefore led to clarify the connections between "trees",
"amalgams" and "SLz". This is the subject of the present work,
which is based on part of a course given at the College de France in
1968/69; the rest of this course has been published elsewhere ([34]).
There are two chapters.
Chapter I begins with the definition of amalgams and the
normal form of their elements. We then pass to trees (§2), and more
precisely to the following question: what can be said of a group G
acting on a tree X when we know the quotient graph G\X as well as
the stabilizers Gx (XE vert X) and Gy (Y E edge X) of the vertices and
edges? We first treat two special cases:
that where G acts freely, i.e. where the G and G reduce to {1};
x y
the group G is then free; this case, which is that of Ihara's theorem,
also gives a simple proof of Schreier's theorem, according to which a
subgroup of a free group is free (§3);
that where the graph G\X is a segment x --y-x', in which case
G may be identified with the amalgam Gx *Gy Gx'; moreover, every
amalgam of two groups is obtained in this way, and uniquely so; one
VIII Introduction
thus gets a convenient equivalence between "amalgams" and
"groups acting on trees with a segment as fundamental domain".
The general case is the object of §5. In the oral course I confined
myself to suggesting its possibility, without verification. The
definitive form of the definitions and theorems, as well as the proofs,
are due to Hyman Bass. The main result says, roughly, that one can
reconstruct G from G\X and the stabilizers G and G it is the
x y:
"fundamental group" of a "graph of groups" carried by G\X;
conversely, every graph of groups is obtained in this way, in an
essentially unique way. Here again, one can give a "normal form"
for the elements of G. The case where G\X is a loop leads to "HNN
groups".
The §6 studies the relations between "amalgams" and "fixed
points". It shows that certain groups, such as SL (Z), SP4(Z), etc.,
3
always have fixed points when they act on trees; this proves that
they are not amalgams. An earlier version of this result has already
been published (Lect. Notes in Math. no. 372, Springer-Verlag,
1974, pp. 633 - 640).
Chapter II begins with the definition and main properties of the
tree X attached to a vector space V of dimension 2 over a local field
K. The vertices of this tree are classes of lattices of V, two lattices
being in the same class if they are homothetic, i.e. if they have the
same stabilizer in GL( V); two vertices are adjacent if they are
represented by nested lattices whose quotient is oflength 1 ; here one
recognizes the notion of "lattice neighbours" which occurs in the
classical definition of the Hecke operator Tp. (For example, when K
is the field Qp and V the Tate module of an elliptic curve E, the
vertices of X correspond to the elliptic curves which are p-isogenic to
E; and one recovers the usual description of p-isogenies by means of
a tree.) Once X is defined, one can apply the results of Chap. I, and
obtain without difficulty the theorem ofIhara cited above, as well as
a theorem of Nagao [19J giving the structure of GL2(k[tJ) as an
amalgam ofGL (k) and the Borel group E(k[tJ). The latter result is
2
generalized in §2 to the following situation: we replace k[tJ by the
affine algebra A of a curve carr = C - {P} with a single point P at
infinity. The group r = GLzCA) then acts on the tree X correspond
ing to the valuation defined by P, and the quotient T\Xhas a simple
interpretation in terms of vector bundles of rank 2 over C. Using
known results on such bundles, one gets structure theorems for r\X,
r;
and hence also for moreover, one obtains information on the
r
homology of and its Euler-Poincare characteristic.
We now mention a few questions, connected with those treated
in the text, on which the reader may usefully consult the
Bibliography:
Jntroduction IX
the theory of ends of discrete groups, and Stallings' theorem ([6],
[7], [8], [30], [35]);
analysis on trees, where the Hecke operator "sum over the
neighbouring vertices" replaces the Laplacian ([13], [14]);
the zetaJunction of a discrete subgroup ofSL (Qp) with compact
2
quotient, cf. [15]; this can be given a simple interpretation in terms
of trees and finite graphs;
r
modular forms relative to the group of Chap. II, §2 and its
congruence subgroups (articles to appear by D. Goss, G. Harder,
W. Li, 1. Weisinger);
Mumford's theory of p-adic Schottky groups and the algebraic
curves with which they are associated ([17], [18]);
cohomological properties of S-arithmetic groups ([28], [29J,
[32], [33], [34]), in particular for function fields (G. Harder,
Invent. Math., 42, 1977, pp. 135 - 175).
This work could not have been done without the friendly and
efficient help of Hyman Bass - in writing as much as in completion
of the results. I have great pleasure in thanking him.
I also thank S. C. Althoen and I. M. Chiswell for the corrections
they have sent me.
Chapter I. Trees and Amalgams
§1 Amalgams
1.1 Direct limits
Let (Gi)iEI be a family of groups, and, for each pair (i,j), let Fij be a set of
homomorphisms of Gi into Gj•
We seek a group G = lim Gi and a family of homomorphisms/;: Gi --+ G such
thatfJ of = /; for allfE Fij, the group and the family being universal in the following
sense:
(*) If H is a group and if hi: Gi --+ H is a family of homomorphisms such that
hj of = hi for allfE Fij, then there is exactly one homomorphism h: G --+ H such that
hi = h o/;. (This amounts to saying Hom(G, H) ~ lim Hom(Gi, H), the inverse limit
being taken relative to the Fij.)
We then say that G is the direct limit of. the Gi, relative to the Fij.
Proposition 1. The pair consisting of G and the family (/;)iEI exists and is unique up to
unique isomorphism.
Uniqueness follows in the usual manner from the universal property (or, what
comes to thHe» . same thing, the fact that G represents the functor H H
lim Hom(G Existence is easy. One can, for example, define G by generators
i,
and relations; one takes the generating family to be the disjoint union of those for
the Gi; as relations, on the one hand the xyz-l where x, y, z belong to the same Gi
andz = xy in Gi, on the other hand the xy-l where xEGi,YE G andy = f(x) for at
j
least one fE Fij•
Example. Take three groups A, G1 and G2 and two homomorphismsfl: A --+ Gb
f2: A --+ Gi. One says that the corresponding group G is obtained by amalgamating
A in G1 and G2 by means offl andf2; we denote it by G1 *A G2. One can have
G = {l} even though fl and f2 are non-trivial (cf. exerc. 2).
Application (Van Kampen Theorem). Let Xbe a topological space covered by two
open sets U1 and U2. Suppose that U U2 and U12 = U1 n U2 are arcwise
b
connected. Let XE U12 be a basepoint. Then the fundamental group 7rl(X; x) is
obtained by taking the three groups 7rl(U1;X), 7rl(U2;X) and 7rl(U12;X) and
amalgamating them according to the homomorphisms
7rl(U12;X)--+7rl(U1;X) and 7rl(U12;X)--+7rl(U2;X).
2 1.1 Amalgams
(For a generalization to the case of a covering by a family of open sets, cf. R.
Crowell, Pac. J. Math. 9, 1959, pp. 43-50.)
Exercises
1) Let/1: A -+ G1 and/2: A -+ G2 be two homomorphisms and let G = G1 *A G2 be the
corresponding amalgam. We define subgroups An, G~ and G~ of A, G1 and G2 recursively by
the following conditions:
Gi
Al = {l}, G~ = {l}, = {I}
G7 = subgroup of Gi generated by fi(An).
Let Aoo, G~ be the unions ofthe An, G7 respectively. Show that}; defines an injection A/A 00 -+
Gi/G~ and that G may be identified with the amalgam of GdG';' and G2/G'; along A/Aoo.
It follows (using the results of no. 1.2) that the kernel of A -+ G is A 00 and that the kernel
of Gi -+ G is G~.
2) Let A = Z, G1 = PSL(2, Q) and G2 = Z/2Z. We take 11: A -+ G1 to be an injection
and 12: A -+ G2 to be a surjection. Show that G1 *A G2 = {l}.
1.2 Structure of amalgams
Suppose we are given a group A, a family of groups (G;)iEI and, for each iEI, an
injective homomorphism A --+ Gi. We identify A with its image in each of the Gi. We
denote by *A G; the direct limit (cf. no. 1.1) of the family (A, G;) with respect to these
homomorphisms, and call it the suml of the G; with A amalgamated.
Example. A = {I}; the corresponding group is denoted * G it is the free product of
i;
the G
i.
We now define the notion of a reduced word. For all i E I we choose a set S; of
right coset representatives of Gi modulo A, and assume 1 E S;; the map (a, s) H as is
then a bijection of A x S; onto G; mapping A x (S; - {I}) onto Gi - A.
Let i = (it. ... , in) be a sequence of elements of I (with n ~ 0) satisfying the
following condition:
(T)
Translator's note: This object is a sum in the sense of category theory, even though it is
1
usually called a "product". We shall avoid calling it either as far as possible, and refer only
to "amalgams", except in the case A = {I}, where the term "free product" is too standard
to be changed.