Related papers: On the model theory of higher rank arithmetic grou…

In this paper, we interpret the theta rank of an irreducible character of a finite classical group in terms of the data from the Lusztig classification. Then we prove the following two results: (1) the agreement of the $U$-rank and the…

We consider a family U of finite universes. The second order quantifier Q_R, means for each u in U quantifying over a set of n(R)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called…

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. In a previous article, we gave a necessary and sufficient condition for X to be free of given rank d…

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$.…

Let $\Gamma$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $\Gamma$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly…

We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of…

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is $\Sigma^0_3$-complete. However, when we consider groups of infinite rank, it becomes $\Sigma^1_1$-complete,…

If $\Gamma$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sym(n) (other than Sym(n) and the alternating group Alt(n)), then the rank of $\Gamma$ is at most $n/2+1$, with finitely many exceptions…

Consider a lattice $\Gamma$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $\Gamma$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its…

Let $\Gamma$ be a finite rank subgroup of $\overline{\mathbb{Q}}^*$. We prove that the multiplicative group of the field generated by all elements in the divisible hull of $\Gamma$, is free abelian modulo this divisible hull. This proves…

We show that many important varieties and sets of varieties of semigroups may be defined by relatively simple and transparent first-order formulas in the lattice of all semigroup varieties.

In a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every…

We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $\Gamma$ is a finitely generated group, then $|G|$ divides $#\Hom(\Gamma,G)$ for every finite group $G$ if and only if $\Gamma$…

The standard interpretation of first-order number theory (PA), according to the generally accepted view, associates well-defined set-theoretic entities with each and every well-formed formula of this system. But this implies that the class…

We prove that any non-cocompact irreducible lattice in a higher rank semi-simple Lie group contains a subgroup of finite index, which has three generators.

A positive integer $m$ will be called a {\it finitistic order} for an element $\gamma$ of a group $\Gamma$ if there exist a finite group $G$ and a homomorphism $h:\Gamma\to G$ such that $h(\gamma)$ has order $m$ in $G$. It is shown that up…

We consider groups defined by non-empty balanced presentations with the property that each relator is of the form R(x,y), where x and y are distinct generators and R(.,.) is determined by some fixed cyclically reduced word R(a,b) that…

Our aim is to prove that if T is a complete first order theory, which is not superstable (no knowledge on this notion is required), included in a theory T_1 then for any lambda > |T_1| there are 2^lambda models of T_1 such that for any two…

Let $G$ be a connected semisimple real algebraic group and $\Gamma<G$ be its Zariski dense discrete subgroup. We prove that if $\Gamma\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $\Gamma$ is virtually a product of…

We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{Sl}(n, \mathbb R)$, $M$ is a compact manifold, and…