Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 

 

Random length-spectrum rigidity for free groups


Author: Ilya Kapovich
Journal: Proc. Amer. Math. Soc. 140 (2012), 1549-1560
MSC (2010): Primary 20Fxx; Secondary 57Mxx, 37Bxx, 37Dxx
Published electronically: September 9, 2011
MathSciNet review: 2869139
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Abstract: We say that a subset $ S\subseteq F_N$ is spectrally rigid if whenever $ T_1, T_2\in \mathrm{cv}_N$ are points of the (unprojectivized) outer space such that $ \vert\vert g\vert\vert _{T_1}=\vert\vert g\vert\vert _{T_2}$ for every $ g\in S$, then $ T_1=T_2$ in $ \mathrm{cv}_N$. It is well known that $ F_N$ itself is spectrally rigid; it also follows from the result of Smillie and Vogtmann that there does not exist a finite spectrally rigid subset of $ F_N$. We prove that if $ A$ is a free basis of $ F_N$ (where $ N\ge 2$), then almost every trajectory of a non-backtracking simple random walk on $ F_N$ with respect to $ A$ is a spectrally rigid subset of $ F_N$.


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Additional Information

Ilya Kapovich
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: kapovich@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11030-X
Received by editor(s): May 21, 2010
Received by editor(s) in revised form: January 24, 2011
Published electronically: September 9, 2011
Additional Notes: The author was supported by NSF grant DMS-0904200
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.