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

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