Title:

Isoperimetric functions for subdirect products and BestvinaBrady groups

In this thesis we investigate the Oehn functions of two different classes of groups: subdirect products,
in particular subdirect products of limit groups; and BestvinaBrady groups.
Let $0 =\Gamma_1 \times \Idots \times \Gamma_n$ be a direct product of $n \geq 3$ finitely
presented groups and let $H$ be a subgroup of $0$. Suppose that each $\Gamma_i$ contains a
finite index subgroup $\Gamma_i' \Ieq \Gamma_i$:'such that the commutator subgroup $[0', 0']$ of
$0' = \Gamma_1' \times \Idots \times \Gamma_n'$ is contained in $H$. Suppose furthermore that,
for each $i$, the subgroup $\Gamma_i H$ has finite index in $0$. We prove that $H$ is finitely
presented and satisfies an isoperimetric inequality given in terms of arearadius pairs for the
$\Gamma_i$ and the dimension of $(O'/H) \otimes \Q$. In the case that each $\Gamma_i$ admits a
polynomialpolynomial arearadius pair, it will follow that $H$ satisfies a polynomial isoperimetric
inequality.
As a corollary we obtain that if $K$ is a subgroup of a direct product of $n$ limit groups and if $K$ is
of type $\textrm{FPL{m}(\Q)$, where $m = \max \{ 2, n1\}$, then $K$ is finitely presented and
.satisfies a polynomial isoperimetric inequality. In particular, we obtain that all finitely presented
subgroups of a direct product of at most $3$ limit groups satisfy a polynomial isoperimetric inequality.
We also prove that if $B$ is a finitely presented BestvinaBrady group, then $B$ admits a quartic
isoperimetric function.
