Finite generating sets of relatively hyperbolic groups and applications to geodesic languages

Abstract

Given a finitely generated relatively hyperbolic group $G$, we construct a finite generating set $X$ of $G$ such that $(G,X)$ has the `falsification by fellow traveler property' provided that the parabolic subgroups $\{H_{\omega }{\}}_{\omega \in \Omega }$ have this property with respect to the generating sets $\{X\cap H_{\omega }{\}}_{\omega \in \Omega }$. This implies that groups hyperbolic relative to virtually abelian subgroups, which include all limit groups and groups acting freely on ${\mathbb{R}}^n$-trees, or geometrically finite hyperbolic groups, have generating sets for which the language of geodesics is regular, and the complete growth series and complete geodesic series are rational. As an application of our techniques, we prove that if each $H_{\omega }$ admits a geodesic biautomatic structure over $X\cap H_{\omega }$, then $G$ has a geodesic biautomatic structure. Similarly, we construct a finite generating set $X$ of $G$ such that $(G,X)$ has the `bounded conjugacy diagrams' property or the `neighbouring shorter conjugate' property if the parabolic subgroups $\{H_{\omega }{\}}_{\omega \in \Omega }$ have this property with respect to the generating sets $\{X\cap H_{\omega }{\}}_{\omega \in \Omega }$. This implies that a group hyperbolic relative to abelian subgroups has a generating set for which its Cayley graph has bounded conjugacy diagrams, a fact we use to give a cubic time algorithm to solve the conjugacy problem. Another corollary of our results is that groups hyperbolic relative to virtually abelian subgroups have a regular language of conjugacy geodesics.

44 pages, 8 figures. Version 3. After comments from the referee added

References in corpus (10)

• Automatic structures, rational growth and geometrically finite hyperbolic groups
• Limit groups and groups acting freely on R^n-trees
• Algorithmic Properties of Relatively Hyperbolic Groups
• The conjugacy problem for relatively hyperbolic groups
• Artin groups of large type are shortlex automatic with regular geodesics
• Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
• Asymptotic dimension of relatively hyperbolic groups
• Finiteness and the falsification by fellow traveler property
• Conjugacy Search Problem for Relatively Hyperbolic Groups
• Conjugacy languages in groups

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• Piecewise excluding geodesic languages
• Negligibility of parabolic elements in relatively hyperbolic groups
• Time complexity of the conjugacy problem in relatively hyperbolic groups