Hyperbolic hydra
We give examples of hyperbolic groups with finite-rank free subgroups of huge (Ackermannian) distortion.
math.GR↗
arXiv subjects
Will Dison
Publications and source records attributed to Will Dison.
Hydra groups
We give examples of CAT(0), biautomatic, free-by-cyclic, one-relator groups which have finite-rank free subgroups of huge (Ackermannian) distortion. This leads to elementary examples of groups whose Dehn functions are similarly extravagant. This behaviour originates in manifestations of Hercules-versus-the-hydra battles in string-rewriting.
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A Subgroup of a Direct Product of Free Groups whose Dehn Function has a Cubic Lower Bound
We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups.
math.GR↗
An Isoperimetric Function for Bestvina-Brady Groups
Given a right-angled Artin group A, the associated Bestvina-Brady group is defined to be the kernel of the homomorphism A \to \mathbb{Z} that maps each generator in the standard presentation of A to a fixed generator of \mathbb{Z}. We prove that the Dehn function of an arbitrary finitely presented Bestvina-Brady group is bounded above by n^4. This is the best possible universal upper bound.
math.GR↗
An isoperimetric function for Stallings' group
We prove that $n^{7/3}$ is an isoperimetric function for a group of Stallings that is finitely presented but not of type $\mathcal{F}_3$. Note: The authors with Robert Young have now proved a quadratic Dehn function for this group. See arXiv:0712.3877
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The Dehn function of Stallings' group
We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F_3 is quadratic. To appear in Geometric and Functional Analysis.
math.GR↗
Isoperimetric functions for subdirect products and Bestvina-Brady groups
In this thesis we investigate the Dehn functions of two different classes of groups: subdirect products, in particular subdirect products of limit groups; and Bestvina-Brady groups. Let D = Γ_1 \times ... \times Γ_n be a direct product of n \geq 3 finitely presented groups and let H be a subgroup of D. Suppose that each Γ_i contains a finite index subgroup Γ_i' \leq Γ_i such that the commutator subgroup [D', D'] of D' = Γ_1' \times ... \times Γ_n' is contained in H. Suppose furthermore that, for each i, the subgroup Γ_i H has finite index in D. We prove that H is finitely presented and satisfies an isoperimetric inequality given in terms of area-radius pairs for the Γ_i and the dimension of (D'/H) \otimes \Q. In the case that each Γ_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 FP_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.
math.GR↗