Eigenvalue gaps for hyperbolic groups and semigroups

Fanny Kassel; Rafael Potrie

doi:10.3934/jmd.2022008

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2022, Volume 18: 161-208. Doi: 10.3934/jmd.2022008

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Eigenvalue gaps for hyperbolic groups and semigroups

Fanny Kassel^1, and
Rafael Potrie^2,

1.
CNRS and Laboratoire Alexander Grothendieck, Institut des Hautes Études Scientifiques, Université Paris-Saclay, 35 route de Chartres, 91440 Bures-sur-Yvette, France
2.
CMAT, Facultad de Ciencias, Universidad de la Repú-blica, Iguá 4225, Montevideo, 11400, Uruguay

Received: February 18, 2020

Revised: August 04, 2021

FK: Partially supported by the Louis D. Foundation. This project received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC starting grant DiGGeS, grant agreement No. 715982).
RP: Partially supported by CSIC-618, FCE-135352, FCE-148740 and MathAmSud. This work was completed while RP was a von Neumann fellow at IAS, funded by Minerva Research Foundation Membership Fund and NSF DMS-1638352.

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Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the $ i^\text{th} $ and $ (i+1)^\text{th} $ Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index $ i $. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.

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Mathematics Subject Classification: Primary: 20M30, 22E40, 37D30; Secondary: 20F67, 37D25.

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