TY - JOUR

T1 - Infinite volume and infinite injectivity radius

AU - Fraczyk, Mikolaj

AU - Gelander, Tsachik

N1 - Funding Information:
Keywords: locally symmetric spaces, discrete subgroups of Lie groups, stationary random subgroups, thin groups, lattices, normal subgroup theorem AMS Classification: Primary: 22E40, 53C24, 22E15, 22F30, 53C35, 32M15. T.G. Supported by ISF-Moked grant 2095/15. © 2023 Department of Mathematics, Princeton University.
Funding Information:
Acknowledgments. We thank Uri Bader for sharing with us some insights concerning stationary measures and Poisson boundaries. Our work was supported by the National Science Foundation under Grant No. DMS-1928930 while the authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020 semester. The second author was partially supported by the Israel Science Foundation grant No. 2919/19.
Publisher Copyright:
© 2023 Department of Mathematics, Princeton University.

PY - 2023

Y1 - 2023

N2 - We prove the following conjecture of Margulis. Let G be a higher rank simple Lie group, and let Λ ≤ G be a discrete subgroup of innite covolume. Then, the locally symmetric space Λ\G/K admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan's property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stuck-Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.

AB - We prove the following conjecture of Margulis. Let G be a higher rank simple Lie group, and let Λ ≤ G be a discrete subgroup of innite covolume. Then, the locally symmetric space Λ\G/K admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan's property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stuck-Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.

KW - discrete subgroups of Lie groups

KW - lattices

KW - locally symmetric spaces

KW - normal subgroup theorem

KW - stationary random subgroups

KW - thin groups

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U2 - 10.4007/annals.2023.197.1.6

DO - 10.4007/annals.2023.197.1.6

M3 - Article

AN - SCOPUS:85146887517

SN - 0003-486X

VL - 197

SP - 389

EP - 421

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -