Filling functions of arithmetic groups

Enrico Leuzinger, Robert Young

Research output: Contribution to journalArticlepeer-review


The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When n is less than the rank of the associated symmetric space, we show that the n-dimensional filling volume function of the lattice grows at the same rate as that of the associated symmetric space, and when n is equal to the rank, we show that the n-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky{Mozes{Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux{Wortman.

Original languageEnglish (US)
Pages (from-to)733-792
Number of pages60
JournalAnnals of Mathematics
Issue number3
StatePublished - May 2021


  • Arithmetic groups
  • Dehn function
  • Filling invariants
  • Probabilistic methods
  • Random ats

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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