Nikolay Nikolov

The rank gradient from a combinatorial viewpoint

This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the amenable case we generalize Lackenby’s trichotomy theorem on finitely presented groups.

Generators and commutators in finite groups; abstract quotients of compact groups

The first part of the paper establishes results about products of commutators in a d-generator finite group G, for example: if H⊲G=〈g1,…,gr〉 then every element of the subgroup [H,G] is a product of f(r) factors of the form

Cartesian products as profinite completions

[h_{1},g_{1}][h_{1}^{prime},g_{1}^{-1}]ldotslbrack h_{r},g_{r}][h_{r}^{prime },g_{r}^{-1}]

Subgroup growth of lattices in semisimple Lie groups

with

Counting primes, groups, and manifolds.

h_{1},h_{1}^{prime},ldots,allowbreak h_{r},h_{r}^{prime }in H

A conjecture on product decompositions in simple groups

. Under certain conditions on H, a similar conclusion holds with the significantly weaker hypothesis that G=H〈g1,…,gr〉, where f(r) is replaced by f1(d,r). The results are applied in the second part of the paper to the study of normal subgroups in finitely generated profinite groups, and in more general compact groups. Results include the characterization of (topologically) finitely generated compact groups which have a countably infinite image, and of those which have a virtually dense normal subgroup of infinite index. As a corollary it is deduced that a compact group cannot have a finitely generated infinite abstract quotient.

Congruence subgroup growth of arithmetic groups in positive characteristic

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups under some natural restrictions.

Lectures on Profinite Topics in Group Theory: An introduction to compact p -adic Lie groups

We give very precise bounds for the congruence subgroup growth of arithmetic groups. This allows us to determine the subgroup growth of irreducible lattices of semisimple Lie groups. In the most general case our results depend on the Generalized Riemann Hypothesis for number fields but we can state the following unconditional theorem: nLet

On finitely generated profinite groups, I: strong completeness and uniform bounds

G

On the growth of L 2-invariants for sequences of lattices in Lie groups

be a simple Lie group of real rank at least 2, different than