Explicit polynomial bounds on Dehn functions of subgroups of hyperbolic groups

TL;DR

This paper provides the first explicit polynomial upper bound on the Dehn function of a finitely presented non-hyperbolic subgroup of a hyperbolic group, specifically bounding it by n^{96}.

Contribution

It establishes a polynomial bound on the Dehn function for Brady's non-hyperbolic subgroup, advancing understanding of subgroup geometry within hyperbolic groups.

Findings

Dehn function of Brady's subgroup bounded by n^{96}

Hyperbolicity constant of the universal cover's 1-skeleton determined

Explicit polynomial bounds on subgroup Dehn functions obtained

Abstract

In 1999 Brady constructed the first example of a non-hyperbolic finitely presented subgroup of a hyperbolic group by fibring a non-positively curved cube complex over the circle. We show that his example has Dehn function bounded above by $n^{96}$. This provides the first explicit polynomial upper bound on the Dehn function of a finitely presented non-hyperbolic subgroup of a hyperbolic group. We also determine the precise hyperbolicity constant for the $1$-skeleton of the universal cover of the cube complex in Brady's construction with respect to the $4$-point condition for hyperbolicity.