Andrei Jaikin-Zapirain

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.

Property (T) for noncommutative universal lattices

We establish a new spectral criterion for Kazhdan’s property (T) which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property (T) for the groups ELn(R), where n≥3 and R is an arbitrary finitely generated associative ring. We also strengthen some of the results on property (T) for Kac-Moody groups from (Dymara and Januszkiewicz in Invent. Math. 150(3):579–627, 2002).

Cohomological goodness and the profinite completion of Bianchi groups

The concept of cohomological goodness was introduced by J-P. Serre in his book on galois cohomology. This property relates the cohomology groups of a group to those of its profinite completion. We develop properties of goodness and establish goodness for certain important groups. We prove for example that the Bianchi groups, that is the groups PSL(2,O) where O is the ring of integers in an imaginary quadratic number field, are good. As an application of our improved understanding of goodness we are able to show that certain natural central extensions of Fuchsian groups are residually finite. A result which contrasts examples of P. Deligne who shows that the analogous central extensions of Sp(4,Z) do not have this property. 2000 Mathematics Subject Classification: Primary 20H05, 11F75; Secondary 14G32, 19B37, 57N10

On Beauville surfaces

We prove that if a finite group G acts freely on a product of two curves C1×C2 so that the quotient S = C1×C2/G is a Beauville surface then C1 and C2 are both non hyperelliptic curves of genus ≥ 6; the lowest bound being achieved when C1 = C2 is the Fermat curve of genus 6 and G = (Z/5Z). We also determine the possible values of the genera of C1 and C2 when G equals S5, PSL2(F7) or any abelian group. Finally, we produce examples of Beauville surfaces in which G is a p-group with p = 2, 3.

On the verbal width of finitely generated pro-

Let p be a prime. It is proved that a non-trivial word w from a free group F has finite width in every finitely generated pro-p group if and only if w 6∈ (F ′)pF ′′. Also it is shown that any word w has finite width in a compact p-adic group.

p

Let G by compact p-adic Lie group and suppose that G is FAb, i.e., that H/(H,H) is finite for every open subgroup H of G. The representation zeta functionG(s) = P �2Irr(G) �(1) s encodes the distribution of continuous irreducible complex characters of G. For p � 3 it is known thatG(s) defines a meromorphic function on C. Wedderburns structure theorem for semisimple algebras implies that �G( 2) = |G| for finite G. We complement this classic result by proving thatG( 2) = 0 for infinite G, assuming p � 3.

groups

Let S be a finite set of powers of p containing 1. It is known that for some choices of S, if P is a finite p-group whose set of character degrees is S, then the nilpotence class of P is bounded by some integer that depends on S, while for some other choices of S such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set S is class bounding if and only if p ¬∈ S. In this article we provide a new approach to this problem. Our main result shows the relevance of certain p-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets S such that p ¬∈ S.

The representation zeta function of a FAb compact p-adic Lie group vanishes at -2

In this work we solve a conjecture of Y. Barnea and M. Isaacs about centralizer sizes and the nilpotency class in nilpotent finite-dimensional Lie algebras and finite p-groups.

Character degrees and nilpotence class of finite -groups: An approach via pro- groups

Abstract In this paper we introduce the concept of weighted deficiency for abstract and pro-p groups and study groups of positive weighted deficiency which generalize Golod–Shafarevich groups. In order to study weighted deficiency, we introduce weighted versions of the notions of rank for groups and index for subgroups and establish weighted analogues of several classical results in combinatorial group theory, including the Schreier index formula. Two main applications of groups of positive weighted deficiency are given. First we construct infinite finitely generated residually finite p-torsion groups in which every finitely generated subgroup is either finite or of finite index—these groups can be thought of as residually finite analogues of Tarski monsters. Second we develop a new method for constructing just-infinite groups (abstract or pro-p) with prescribed properties; in particular, we show that graded group algebras of just-infinite groups can have exponential growth. We also prove that every group of positive weighted deficiency has a hereditarily just-infinite quotient. This disproves a conjecture of Boston on the structure of quotients of certain Galois groups and solves Problem 15.18 from the Kourovka notebook.

Centralizer sizes and nilpotency class in Lie algebras and finite p-groups

In this paper we prove that any finite group of rankr, with an automorphism whose centralizer hasm points, has a characteristic soluble subgroup of (m, r)-bounded index andr-bounded derived length. This result gives a positive answer to a problem raised by E. I. Khukhro and A. Shalev (see also Problem 13.56 from the “Kourovka Notebook” [Kou]).