Related papers: Finitely presented groups with transcendental spec…

We give a general asymptotic formula for the growth rate of the number of indecomposable summands in the tensor powers of representations of finite groups, over a field of arbitrary characteristic. In characteristic zero we obtain…

Necessary and sufficient conditions for a group to possess a faithful irreducible representation are investigated

Let $d \geq 2$ be an integer. We conjecture that there is a finitely generated perfect group whose homomorphic images include all finite $d$-generated perfect groups. We prove a special case of this conjecture for the finite perfect groups…

We study some spectral properties of random walks on infinite countable amenable groups with an emphasis on locally finite groups, e.g. the infinite symmetric group. On locally finite groups, the random walks under consideration are driven…

We consider deformations of bounded complexes of modules for a profinite group G over a field of positive characteristic. We prove a finiteness theorem which provides some sufficient conditions for the versal deformation of such a complex…

In this paper, using some properties of fundamental groups and covering spaces of connected polyhedra and CW-complexes, we present topological proof for some famous theorems about finitely presented groups.

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are…

We construct a new large family of finitely generated groups with continuum many values of the following monotone parameters: spectral radius, critical percolation, and asymptotic entropy. We also present several open problems on other…

A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all…

We construct a finitely presented group with property (T) which can not act on on reasonable spaces. Such group is constructed using an generalization of Hall embedding theorem, where property (T) is added at the expense of weakening the…

We extend some results of Gun, Murty, and Rath on elliptic modular forms. We take ANY Fuchsian triangle group with a cusp and look at power series expansions in a natural parameter around that cusp. Consider the automorphic forms for such a…

We construct the first examples of residually finite non-exact groups. The construction is based on author's earlier construction of groups containing isometrically expanders using a graphical small cancellation.

We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group $\Gamma := \mathbf{SL}_2(A)$, where $A := \mathbb{F}_q[X,X^{-1}]$ is the ring of Laurent polynomials with…

We introduce a model for random groups in varieties of $n$-periodic groups as $n$-periodic quotients of triangular random groups. We show that for an explicit $d_{\mathrm{crit}}\in(1/3,1/2)$, for densities $d\in(1/3,d_{\mathrm{crit}})$ and…

The paper contains an exposition of recent as well as old enough results on determinantal random point fields. We start with some general theorems including the proofs of the necessary and sufficient condition for the existence of the…

Asymptotic properties of finitely generated subgroups of free groups, and of finite group presentations, can be considered in several fashions, depending on the way these objects are represented and on the distribution assumed on these…

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $\Gamma$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type…

We construct the first examples of infinite sharply 2-transitive groups which are finitely generated. Moreover, we construct such a group that has Kazhdan property (T), is simple, has exactly four conjugacy classes, and we show that this…

This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of…

We introduce a new real valued invariant for finitely presented groups called residual deficiency. Its main property is the following. Let G be a finitely presented group. If the residual deficiency of G is greater than one, then G has a…