Let G be a finite product of SL(2,Ki)′s for local fields Ki of characteristic zero. We present a discreteness criterion for non-solvable subgroups of G containing an irreducible lattice of a maximal unipotent subgroup of G. In particular such a subgroup has to be arithmetic. This extends a previous result of A. Selberg when G is a product of SL2(R)′s.

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a no...

A. Smoktunowicz and L. Vendramin conjectured that if A is a finite skew brace with solvable additive group, then the multiplicative group of solvable. In this short note we make step towards positive solution conjecture proving minimal non-solvable not simple. On way to obtaining result, prove correct in case when order divisible by 3.

We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a connected non-solvable Lie group H contains a dense free subgroup of rank ≤ 2 · dimH . As an application, we o...

According to Thurston’s stability theorem, every group of C diffeomorphisms of the closed interval is locally indicable (i.e., every finitely generated subgroup factors through Z). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the group F2 ⋉ Z , although locally indicable, does not embed into Diff1+(]0, 1[). (Here F2 i...

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an ominimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R ·S where R is the solvable radical of G. We also prove that any semisimple subalgebra of t...

We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we show that the algebraic entropy of discrete subgroups of a given Lie group is uniformly bounded away from zero. In this note, we summarize some results whose f...

Let G be a group, and $${{\,\mathrm{Sol}\,}}(G)=\{x \in : \langle x,y \rangle \text { is solvable for all } y G\}$$ . We associate graph $$\mathcal {NS}_G$$ (called the non-solvable of G) with whose vertex set $$G \setminus {{\,\mathrm{Sol}\,}}(G)$$ two distinct vertices are adjacent if they generate subgroup. In this paper, we study many properties particular, obtain results on degree, cardina...

We prove that if L = lim ←−Ln (n ∈ N), where each Ln is a finite dimensional semisimple Lie algebra, and A is a finite codimensional ideal of L, then L/A is also semisimple. We show also that every finite dimensional homomorphic image of the cartesian product of solvable (nilpotent) finite dimensional Lie algebras is solvable (nilpotent). Mathematics Subject Classification: 14L, 16W, 17B45