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


Authors: Yago Antolín and Laura Ciobanu
Journal: Trans. Amer. Math. Soc. 368 (2016), 7965-8010
MSC (2010): Primary 20F65, 20F10, 20F67, 68Q45
DOI: https://doi.org/10.1090/tran/6701
Published electronically: March 21, 2016
MathSciNet review: 3546790
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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 `neighboring 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.


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Additional Information

Yago Antolín
Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nash- ville, Tennessee 37240
Email: yago.anpi@gmail.com

Laura Ciobanu
Affiliation: Institut de Mathématiques, University of Neuchâtel, R. Emile-Argand 11, CH-2000 Neuchâtel, Switzerland
Email: laura.ciobanu@unine.ch

DOI: https://doi.org/10.1090/tran/6701
Keywords: Relatively hyperbolic groups, Cayley graphs, growth series, conjugacy problem, languages of geodesics, falsification by fellow traveler property, bounded conjugacy diagrams, (bi)automatic groups, rational growth
Received by editor(s): November 6, 2014
Received by editor(s) in revised form: February 4, 2015, and March 3, 2015
Published electronically: March 21, 2016
Article copyright: © Copyright 2016 American Mathematical Society