We will describe a collection of simple geometric

constructions which yield groups with interesting first and

second order Dehn functions. The second order Dehn function is a

higher dimensional analogue of the standard Dehn function (or isoperimetric function).

A group G is VF if it has a finite-index subgroup

admitting a finite classifying space. K. S. Brown showed that

each VF group contains only finitely many conjugacy classes of

elements of prime power order. I shall give a construction of

a VF group containing infinitely many conjugacy classes of

elements of order 6. This is based on earlier joint work of

mine with B. E. A. Nucinkis.

fields

In this talk we'll focus on proofs of Nagao's theorem that SL_2(F[t])

is not finitely generated, and of Behr's theorem that SL_3(F[t]) is not

finitely presented. Here F[t] is a ring of polynomials in one variable t

with coefficients in a finite field F.

The proofs I will present are special cases of the proof Kai-Uwe Bux and I

used to verify the conjecture that an arithmetic subgroup of a reductive

group G defined over a global function field K is of type FP_\infty if

and only if the semisimple K-rank of G equals 0.

This conjecture had its roots in the work of Serre and Stuhler. Our proof

is motivated by the Epstein-Thurston proof that SL_n(Z) is not combable

when n>2.