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Linear actions of free groups. (English) Zbl 0967.37016

Given a discrete subgroup \(\Gamma\) of \(SL(d,{\mathbb R})\) and a fixed non zero vector, it is interesting to consider the orbits \(\{Av, A \in \Gamma\}\). For groups which are lattices, the set \(\{A \in \Gamma\), \(\|Av \|\leq T\}\) is infinite. For other groups, this set may be finite for certain choices of \(v\). The question is how this counting function behaves as \(T\) tends to infinity.
The authors consider the case where \(\Gamma\) is a free group which satisfies two generic conditions. They get an estimate of the cardinality of \(\{A^l \in \Gamma\), \(\|Av\|\leq T\}\) for all sufficiently large \(l\): the counting function is equivalent with \(C T^p\), where \(p\) is the abscissa of convergence of a Dirichlet series. The proof is based first on the study of the linear action of the matrices on the projective space, secondly on thermodynamic formalism applied to a Poincaré series.
The authors also describe the distribution of the orbits \(\{A^l v\), \(A \in \Gamma\}\) on the projective space. Their result could be viewed as an analogue of the Patterson-Sullivan measure for a hyperbolic manifold.

MSC:

37C35 Orbit growth in dynamical systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
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