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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
##### Keywords:
loxodromic subgroups; special linear group
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