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Title: Isoperimetric functions for subdirect products and Bestvina-Brady groups
Author: Dison, William John
ISNI:       0000 0001 3425 2650
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2008
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In this thesis we investigate the Oehn functions of two different classes of groups: subdirect products, in particular subdirect products of limit groups; and Bestvina-Brady 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 area-radius pairs for the $\Gamma_i$ and the dimension of $(O'/H) \otimes \Q$. In the case that each $\Gamma_i$ admits a polynomial-polynomial area-radius 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 $m = \max \{ 2, n-1\}$, 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 Bestvina-Brady group, then $B$ admits a quartic isoperimetric function.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available