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.

中文翻译：

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3空间中2格空间的动力学

我们研究\（{{\ rm SL_3}（\ mathbb {R}）} \）及其子群在均质空间X上的动力学，该均质空间X由\（{\ mathbb {R} ^ 3} \）。我们关注的情况是，行动小组在\（{{\ rm SL_3}（\ mathbb {R}）} \）或\（{{\ rm SO（2,1）}（\ mathbb { R}）} \）。使用Benoist和Quint的技术，我们证明了对于\（{{\ rm SL_3}（\ mathbb {R}）} \）上受紧支持的概率测度\（{\ mu} \），其支持产生了一个Zariski组\（{{\ rm SL_3}（\ mathbb {R}）} \）中密集，则存在唯一的\（{\ mu} \）平稳概率测度X。当Zariski闭为\（{{\ rm SO（2,1）}（\ mathbb {R}）} \）时，我们就平稳测度建立了一定的二分法，并发现了一个令人惊讶的现象：泊松边界可以嵌入X。嵌入具有代数性质，并提出了许多自然的开放性问题。此外，讨论了激励应用于数字几何中的问题。