Infinite volume and infinite injectivity radius

Mikolaj Fraczyk*, Tsachik Gelander

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)389-421
Number of pages33
JournalAnnals of Mathematics
Issue number1
StatePublished - 2023
Externally publishedYes


  • discrete subgroups of Lie groups
  • lattices
  • locally symmetric spaces
  • normal subgroup theorem
  • stationary random subgroups
  • thin groups

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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