The Hopf–Tsuji–Sullivan dichotomy in higher rank and applications to Anosov subgroups

Marc Burger; Or Landesberg; Minju Lee; Hee Oh

doi:10.3934/jmd.2023008

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2023, Volume 19: 301-330. Doi: 10.3934/jmd.2023008

This volume Previous Article The limit set of non-orientable mapping class groups Next Article Horospherical invariant measures and a rank dichotomy for Anosov groups

The Hopf–Tsuji–Sullivan dichotomy in higher rank and applications to Anosov subgroups

1.
Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
2.
Department of Mathematics, Yale University, New Haven, CT 06520, USA

MINJU LEE
MINJU LEE

Received: August 9, 2021

Revised: October 31, 2022

Published online: May 4, 2023

HO: Partially supported by the NSF.

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Abstract

We establish an extension of the Hopf–Tsuji–Sullivan dichotomy to any Zariski dense discrete subgroup of a semisimple real algebraic group $ G $. We then apply this dichotomy to Anosov subgroups of $ G $, which surprisingly presents a different phenomenon depending on the rank of the ambient group $ G $.

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Mathematics Subject Classification: Primary: 37A17, 37A40, 22E40; Secondary: 22F30.

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References

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