Contemporary Mathematics
Volume 188, 1995
Discrete Groups, Grothendieck Rings and
Families of Finite Subgroups
ALEJANDRO ADEM
Dedicated to Samuel Gitler on occasion of his sixtieth birthday.
0. Introduction
Understanding the representation theory of an infinite discrete group is usually
very difficult. Nevertheless, there are certain classes of discrete groups (such as
arithmetic groups) which satisfy algebraic properties similar to those of a finite
group. For example, an arithmetic group has only a finite number of elements
of finite order up to conjugacy, and the same is true for other large classes of
groups such as the mapping class groups, the outer automorphism group of a
free group, hyperbolic groups, etc.
In this paper we look at the representation theory which is determined by
finite subgroups.
More precisely, if r is a discrete group with a finite number of
finite subgroups up to conjugacy (the FCP condition) we introduce
DEFINITION
If
:F
is a family of finite subgroups, and R is a commutative
ring, then
C.r(R[f])
=
lim
G(R[H]),
HE.F
where G(R[H]) denotes the usual Grothendieck ring of finitely generated
R[Hj-
modules.
After providing preliminaries in section 1, we prove some basic facts about
the rings C.r(R[f]) which follow easily from the corresponding results for finite
groups. We summarize them below (notation as above):
(1)
If
Sis a commutative ring with 1 satisfying the minimal conditions on ideals
then C.r(S[r]) is free abelian of finite rank.
1991 Mathematics Subject Classification. 20C07, 55R40, 19A22.
Partially supported by an NSF grant, an NSF Young Investigator Award and the ETH-
Ziirich.
This paper is in final form and no version of it will be submitted for publication elsewhere.
©
1995 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/188/02228
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