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

Abstract

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 {Hω}ω∈Ω have this property with respect to the generating sets {X ∩ Hω}ω∈Ω. This implies that groups hyperbolic relative to virtually abelian subgroups, which include all limit groups and groups acting freely on 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 Hω admits a geodesic biautomatic structure over X ∩ Hω, 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 {Hω}ω∈Ω have this property with respect to the generating sets {X ∩Hω}ω∈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.

Geometry of the word problem for 3-manifold groups

“…Background and details on relatively hyperbolic groups used in this paper can be found in [25,13,20,1].…”

“…Antolin and Ciobanu studied geodesics and language theoretic properties of relatively hyperbolic groups in [1]. For example, they show that every finitely generated relatively hyperbolic group has a finite generating set, A, such that each Γ A∩H j (H j ) isometrically embeds in Γ A (G) [1, Lemma 5.3] (in fact every finite generating set can be extended to one with this property).…”

“…In comparison, finite convergent rewriting systems have been found for all fundamental groups of Seifert fibered knot complements, namely the torus knot groups, by Dekov [11], and for fundamental groups of alternating knot complements, by Chouraqui [10]. In the case of a finite volume hyperbolic 3-manifold M , the fundamental group π 1 (M ) is hyperbolic relative to the collection of fundamental groups of its torus boundary components by a result of Farb [13], and so by closure of the class of (prefix-closed) biautomatic groups with respect to relative hyperbolicity (shown by Rebbecchi in [27]; see also [1]), the group π 1 (M ) is biautomatic.…”

The generalised word problem in hyperbolic and relatively hyperbolic groups

“…This article is structured as follows. In Section 2, we summarise the basic properties of relatively hyperbolic groups that we shall need, and we recall some of their properties that are proved in [3]. In Section 3, we introduce the concept of extended Dehn algorithms for solving the word and generalised word problems in groups and we recall some results pertaining to relatively hyperbolic groups that are proved in [9].…”

“…We shall need to use two more properties of relatively hyperbolic groups that are proved in [3]. Now suppose that v is any word in X * .…”