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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.
MSC:
14L35 Classical groups (algebro-geometric aspects)
20G30 Linear algebraic groups over global fields and their integers
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