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

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## Abstract

We study the dynamics of \({{\rm SL_3}(\mathbb{R})}\) and its subgroups on the homogeneous space *X* consisting of homothety classes of rank-2 discrete subgroups of \({\mathbb{R}^3}\). We focus on the case where the acting group is Zariski dense in either \({{\rm SL_3}(\mathbb{R})}\) or \({{\rm SO(2,1)}(\mathbb{R})}\). Using techniques of Benoist and Quint we prove that for a compactly supported probability measure \({\mu}\) on \({{\rm SL_3}(\mathbb{R})}\) whose support generates a group which is Zariski dense in \({{\rm SL_3}(\mathbb{R})}\), there exists a unique \({\mu}\)-stationary probability measure on *X*. When the Zariski closure is \({{\rm SO(2,1)}(\mathbb{R})}\) 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.

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## Notes

### Acknowledgements

We would like to express our gratitude to Elon Lindenstrauss for correcting a mistake in an earlier draft. We would also like to thank Uri Bader, Yves Benoist, Alex Eskin, Alex Furman, Elon Lindenstrauss, Amos Nevo, Jean-François Quint, Ron Rosenthal, Nicolas de Saxcé, Barak Weiss and Cheng Zheng for their support encouragement and assistance. We acknowledge the support of ISF grant 357/13 and the warm hospitality and splendid environment provided by MSRI where some of the research was conducted during the special semester Geometric and Arithmetic Aspects of Homogeneous Dynamics held on 2015. We also acknowledge the partial support of ISF grant 871/17. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 754475).

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