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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
DOI: https://doi.org/10.1090/tran/6933
Published electronically: June 13, 2017
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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
Email: thomas.koberda@gmail.com

Johanna Mangahas
Affiliation: Department of Mathematics, 244 Mathematics Building, University at Buffalo, Buffalo, New York 14260
Email: mangahas@buffalo.edu

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
Email: samuel.taylor@temple.edu

DOI: https://doi.org/10.1090/tran/6933
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