The distortion dimension of Q -rank 1 lattices

Enrico Leuzinger, Robert Young

Research output: Contribution to journalArticle

Abstract

Let X= G/ K be a symmetric space of noncompact type and rank k≥ 2. We prove that horospheres in X are Lipschitz (k- 2) -connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible Q-rank-1 lattice Γ in a linear, semisimple Lie group G of R-rank k is k- 1. That is, given m< k- 1 , a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) Γ , and a (m+ 1) -ball B in X (or G) filling S, there is a (m+ 1) -ball B in Γ filling S such that vol B∼ vol B. In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension k- 1.

Original languageEnglish (US)
Pages (from-to)69-87
Number of pages19
JournalGeometriae Dedicata
Volume187
Issue number1
DOIs
StatePublished - Apr 1 2017

Keywords

  • Arithmetic groups
  • Dehn functions
  • Horospheres
  • Lipschitz connectivity
  • Subgroup distortion
  • Symmetric spaces

ASJC Scopus subject areas

  • Geometry and Topology

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