Dynamics on the space of 2-lattices in 3-space

Oliver Sargent, Uri Shapira

Research output: Contribution to journalArticlepeer-review


We study the dynamics of SL 3(R) and its subgroups on the homogeneous space X consisting of homothety classes of rank-2 discrete subgroups of R3. We focus on the case where the acting group is Zariski dense in either SL 3(R) or SO (2 , 1) (R). Using techniques of Benoist and Quint we prove that for a compactly supported probability measure μ on SL 3(R) whose support generates a group which is Zariski dense in SL 3(R) , there exists a unique μ-stationary probability measure on X. When the Zariski closure is SO (2 , 1) (R) we establish a certain dichotomy regarding stationary measures and discover a surprising phenomenon: The Poisson boundary can be embedded in X. The embedding is of algebraic nature and raises many natural open problems. Furthermore, motivating applications to questions in the geometry of numbers are discussed.

Original languageEnglish
Pages (from-to)890-948
Number of pages59
JournalGeometric and Functional Analysis
Issue number3
StatePublished - 1 Jun 2019

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology


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