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Equidistribution of dense subgroups on nilpotent Lie groups. (English) Zbl 1194.22012
The author considers a nilpotent Lie group \(\Gamma\) generated by a finite set \(S\) and the \(n\)-ball \(S^n\) for the word metric induced by \(S\) on \(\Gamma\). In this article, he proves that if \(\Gamma\) is a dense subset of a simply connected nilpotent Lie group, then \(S^n\) becomes equidistributed on the nilpotent Lie group as \(n\) tends to infinity. More precisely the main theorem of this paper states the following: if \(G\) is a closed subgroup of a simply connected nilpotent Lie group and if \(\phi:\Gamma\rightarrow G\) is a homomorphism with dense image, then there exists a constant \(C>0\) such that for every bounded Borel subset \(B\subset G\) with negligible boundary, \[ \lim_{n\rightarrow \infty}\frac{|S^n\cap \phi^{-1}(B)|}{n^{d(\Gamma)-d(G)}}= C \cdot \text{vol}_G(B)\;; \] here \(d(\Gamma)\) and \(d(G)\) denote the homogeneous dimension at infinity of \(\Gamma\) and \(G\) respectively.
The introduction starts with a brief outline of questions and results about ratio limit theorems as motivation for the article. Then the main result of the paper (given above) is stated together with a corollary. The second section is devoted to present some background on quasinorms and homogeneous structures associated to a nilpotent Lie group, as well as to define what the author calls “nicely growing sets”. In the third section the main result is proved using Malcev’s rigidity property and ergodic theory.
As a corollary the author combines his result with an article of Alexopoulos; he obtains a local limit theorem for a probability measure on a simply connected nilpotent Lie group such that the support of the measure is finitely generated and generates a dense subgroup.

22E40 Discrete subgroups of Lie groups
22E25 Nilpotent and solvable Lie groups
22D40 Ergodic theory on groups
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A80 Analysis on other specific Lie groups
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