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\(H\)-loxodromic subgroups. (Sous-groupes \(H\)-loxodromiques.) (English. French summary) Zbl 1220.14031
Let \(p>0\) be prime, and let \(k\) be a finite extension of \(\mathbb{Q}_{p}.\) Let \(H\) be a subgroup of finite index in \(k^{\ast}\). Endowing \(G:=\text{SL}\left( n,k\right)\) with the Zariski topology as a \(\mathbb{Q}_{p}\)-group, one may ask: under what conditions does there exist a Zariski-dense subgroup of \(G\) such that \(H\) contains the eigenvalues of each element of \(G\)? This can be seen as a non-archimedean analogue to the results in [Y. Benoist, Invent. Math. 141, No. 1, 149–193 (2000; Zbl 0957.22008)], where it is shown that \(\text{SL}\left( n,\mathbb{R}\right) \) has a subgroup which is Zariski-dense with \(H=\left( \mathbb{R}^{+}\right) ^{\ast }\) if and only if \(n\) is not congruent to \(2\text{ mod }4\).
In the work under review, a straightforward answer to this question is given, namely, such a subgroup exists if and only if either \(-1\in H\) or \(n\) is not congruent to \(2\text{ mod }4\). The key to this construction is the study of loxodromic elements, that is, elements whose eigenvalues are all distinct in valuation. The subgroup consists of diagonalizable loxodromic matrices.
14L35 Classical groups (algebro-geometric aspects)
20G30 Linear algebraic groups over global fields and their integers
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