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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The geometry of purely loxodromic subgroups of right-angled Artin groups

Authors: Thomas Koberda, Johanna Mangahas and Samuel J. Taylor
Journal: Trans. Amer. Math. Soc. 369 (2017), 8179-8208
MSC (2010): Primary 20F36; Secondary 57M07
Published electronically: June 13, 2017
MathSciNet review: 3695858
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Abstract: We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $ A(\Gamma )$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $ \mathrm {Mod}(S)$. In particular, such subgroups are quasiconvex in $ A(\Gamma )$. In addition, we identify a milder condition for a finitely generated subgroup of $ A(\Gamma )$ that guarantees it is free, undistorted, and retains finite generation when intersected with $ A(\Lambda )$ for subgraphs $ \Lambda $ of $ \Gamma $. These results have applications to both the study of convex cocompactness in $ \mathrm {Mod}(S)$ and the way in which certain groups can embed in right-angled Artin groups.

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Additional Information

Thomas Koberda
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904-4137

Johanna Mangahas
Affiliation: Department of Mathematics, 244 Mathematics Building, University at Buffalo, Buffalo, New York 14260

Samuel J. Taylor
Affiliation: Department of Mathematics, 10 Hillhouse Ave, Yale University, New Haven, Connecticut 06520
Address at time of publication: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Keywords: Right-angled Artin group, extension graph, convex cocompact subgroup, loxodromic isometry
Received by editor(s): January 5, 2015
Received by editor(s) in revised form: January 27, 2016, and March 8, 2016
Published electronically: June 13, 2017
Article copyright: © Copyright 2017 American Mathematical Society