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

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 $\{H_ω\}_{ω\in Ω}$ have this property with respect to the generating sets $\{X\cap H_ω\}_{ω\in Ω}$. 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 $H_ω$ admits a geodesic biautomatic structure over $X\cap H_ω$, 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 $\{H_ω\}_{ω\in Ω}$ have this property with respect to the generating sets $\{X\cap H_ω\}_{ω\in Ω}$. 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.

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