Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

Aaron Brown; David Fisher; Sebastian Hurtado

doi:10.4007/annals.2022.196.3.1

Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

Aaron Brown^*, David Fisher, Sebastian Hurtado

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if is a cocompact lattice in SL(n;R), M is a compact manifold, and ! a volume form on M, we show that any homomorphism (Formula Presented)) has finite image if the dimension of M is less than n - 1 and that any homo-morphism (Formula Presented) has finite image if the dimension of M is less than n. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Laorgue's strong property (T) to establish the existence of an invariant Riemannian metric.

Original language	English (US)
Pages (from-to)	891-940
Number of pages	50
Journal	Annals of Mathematics
Volume	196
Issue number	3
DOIs	https://doi.org/10.4007/annals.2022.196.3.1
State	Published - Nov 2022

Funding

D.F. was partially supported by NSF Grant DMS-1308291. This work was begun while he visited Chicago, a visit partially supported by NSF RTG Grant DMS-1344997. Travel for A.B. was supported by NSF grants DMS 1107452, 1107263, 1107367, “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network). A.B. was partially supported by NSF grant DMS-2020013. S.H. was partially supported by the Sloan Fellowship Foundation. © 2022 Department of Mathematics, Princeton University.

Keywords

Actions of abelian groups
Actions of lattices
Lattices in semisimple lie groups
Lyapunov exponents
Measure rigidity
Property (t)
Ratner theory
Zimmer program

ASJC Scopus subject areas

Mathematics (miscellaneous)

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10.4007/annals.2022.196.3.1

Cite this

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title = "Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)",

abstract = "We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if is a cocompact lattice in SL(n;R), M is a compact manifold, and ! a volume form on M, we show that any homomorphism (Formula Presented)) has finite image if the dimension of M is less than n - 1 and that any homo-morphism (Formula Presented) has finite image if the dimension of M is less than n. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Laorgue's strong property (T) to establish the existence of an invariant Riemannian metric.",

keywords = "Actions of abelian groups, Actions of lattices, Lattices in semisimple lie groups, Lyapunov exponents, Measure rigidity, Property (t), Ratner theory, Zimmer program",

author = "Aaron Brown and David Fisher and Sebastian Hurtado",

note = "Funding Information: D.F. was partially supported by NSF Grant DMS-1308291. This work was begun while he visited Chicago, a visit partially supported by NSF RTG Grant DMS-1344997. Travel for A.B. was supported by NSF grants DMS 1107452, 1107263, 1107367, “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network). A.B. was partially supported by NSF grant DMS-2020013. S.H. was partially supported by the Sloan Fellowship Foundation. {\textcopyright} 2022 Department of Mathematics, Princeton University. Publisher Copyright: {\textcopyright} 2022 Department of Mathematics, Princeton University.",

year = "2022",

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doi = "10.4007/annals.2022.196.3.1",

language = "English (US)",

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AU - Brown, Aaron

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N1 - Funding Information: D.F. was partially supported by NSF Grant DMS-1308291. This work was begun while he visited Chicago, a visit partially supported by NSF RTG Grant DMS-1344997. Travel for A.B. was supported by NSF grants DMS 1107452, 1107263, 1107367, “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network). A.B. was partially supported by NSF grant DMS-2020013. S.H. was partially supported by the Sloan Fellowship Foundation. © 2022 Department of Mathematics, Princeton University. Publisher Copyright: © 2022 Department of Mathematics, Princeton University.

PY - 2022/11

Y1 - 2022/11

N2 - We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if is a cocompact lattice in SL(n;R), M is a compact manifold, and ! a volume form on M, we show that any homomorphism (Formula Presented)) has finite image if the dimension of M is less than n - 1 and that any homo-morphism (Formula Presented) has finite image if the dimension of M is less than n. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Laorgue's strong property (T) to establish the existence of an invariant Riemannian metric.

AB - We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if is a cocompact lattice in SL(n;R), M is a compact manifold, and ! a volume form on M, we show that any homomorphism (Formula Presented)) has finite image if the dimension of M is less than n - 1 and that any homo-morphism (Formula Presented) has finite image if the dimension of M is less than n. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Laorgue's strong property (T) to establish the existence of an invariant Riemannian metric.

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KW - Lyapunov exponents

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KW - Ratner theory

KW - Zimmer program

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Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

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