The Hanna Neumann conjecture for Demushkin Groups (with Mark Shusterman)

Advances in Mathematics 349 (2019), 1-28. (Demushkin.pdf)

We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups of a nonsolvable Demushkin group.

back

The strong Atiyah and Lück approximation conjecture for one-relator groups (with Diego López-Álvarez)

Mathematische
Annalen (2019), 1-53. (onerelator.pdf)

L2-Betti numbers and their analogues in positive characteristic

Groups St Andrews 2017 in Birmingham, 346-406, London Math. Soc. Lecture Note Ser., 455, Cambridge Univ. Press, Cambridge, 2019. (surveyl2.pdf)

In this article, we give a survey of results on L2-Betti numbers and their analogues in positive characteristic. The main emphasis is made on the Lück approximation conjecture and the strong Atiyah conjecture.

An infinite compact Hausdorff group has uncountably many conjugacy classes (with N. Nikolov)

Proc. of the AMS, 147 (2019), 4083-4089 (conjcompact.pdf)

back

Recognition of being fibered for compact 3-manifolds,

Geometry and Topology (2019), 1-11 (fibering.pdf)

Let M be a compact orientable 3-manifold. We show that if the profinite completion of the fundamental group of M is isomorphic to the profinite completion of a free-by-cyclic group or to the profinite completion of a surface-by-cyclic group, then M fibres over the circle with compact fibre.

The base change in the Atiyah and the Lück approximation conjectures

Geom. Funct. Anal. 29 (2019), 464-538. (sac.pdf)

back

Finite 2-groups with odd number of conjugacy classes (with J. Tent)

Trans. Amer. Math. Soc. 370 (2018), no. 5, 3663–3688. , arxiv version

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if k is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly k real conjugacy classes. On the other hand we construct infinitely many finite 2-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-p techniques and, in particular, we use the Kneser classification of semi-simple p-adic algebraic groups.

back

Units of group rings, the Bogomolov multiplier, and the fake degree conjecture (with Javier García-Rodríguez and Urban Jezernik)

(modular_units.pdf) Mathematical Proceedings of the Cambridge Philosophical Society, DOI: https://doi.org/10.1017/S0305004116000748

Groups graded by root systems and property (T) (with Mikhail Ershov, Martin Kassabov and Zezhou Zhang)

**On the
number of conjugacy classes of finite nilpotent groups**

**On
the
structure of normal subgroups of potent p-groups (with **J.
González-Sánchez**)**

**Weak graded analogues of
Gauss lemma and Eisenstein criterion.**

__Home__**
| **__University__**
| **__Math.
Dep.__

It is shown that the strong Atiyah
conjecture and the Lück approximation conjecture hold for locally
indicable groups. In particular, this implies that one-relator groups
satisfy the strong Atiyah conjecture. We also show that the center
conjecture, the independence conjecture and the strong eigenvalue
conjecture hold for these groups.

As a byproduct we prove that the group
algebra of a locally indicable group over a field of characteristic zero
has a Hughes-free epic division algebra and, in particular, it is
embedded in a division algebra.

L2-Betti numbers and their analogues in positive characteristic

Groups St Andrews 2017 in Birmingham, 346-406, London Math. Soc. Lecture Note Ser., 455, Cambridge Univ. Press, Cambridge, 2019. (surveyl2.pdf)

In this article, we give a survey of results on L2-Betti numbers and their analogues in positive characteristic. The main emphasis is made on the Lück approximation conjecture and the strong Atiyah conjecture.

An infinite compact Hausdorff group has uncountably many conjugacy classes (with N. Nikolov)

Proc. of the AMS, 147 (2019), 4083-4089 (conjcompact.pdf)

back

Recognition of being fibered for compact 3-manifolds,

Geometry and Topology (2019), 1-11 (fibering.pdf)

Let M be a compact orientable 3-manifold. We show that if the profinite completion of the fundamental group of M is isomorphic to the profinite completion of a free-by-cyclic group or to the profinite completion of a surface-by-cyclic group, then M fibres over the circle with compact fibre.

The base change in the Atiyah and the Lück approximation conjectures

Geom. Funct. Anal. 29 (2019), 464-538. (sac.pdf)

In this paper we prove the general Lück
approximation conjecture for sofic groups over an arbitrary field of
zero characteristic. As a corollary we obtain that if the strong Atiyah
conjecture holds for a sofic group G over the field of algebraic
numbers, then it also holds for G over the field of complex numbers.
Among other consequences we obtain that a strong version of the
algebraic eigenvalue conjecture, the center conjecture and the
independence conjecture hold for sofic groups.

back

Finite 2-groups with odd number of conjugacy classes (with J. Tent)

Trans. Amer. Math. Soc. 370 (2018), no. 5, 3663–3688. , arxiv version

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if k is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly k real conjugacy classes. On the other hand we construct infinitely many finite 2-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-p techniques and, in particular, we use the Kneser classification of semi-simple p-adic algebraic groups.

back

Units of group rings, the Bogomolov multiplier, and the fake degree conjecture (with Javier García-Rodríguez and Urban Jezernik)

(modular_units.pdf) Mathematical Proceedings of the Cambridge Philosophical Society, DOI: https://doi.org/10.1017/S0305004116000748

Approximation by subgroups of finite index and the Hanna Neumann conjecture

Duke Math. J., 166(2017), 1955-1987. (hannaneumann.pdf)We establish the Strengthened Hanna Neumann conjecture for pro-p groups and present a new proof of the original Strengthened Hanna Neumann conjecture for abstract groups.

Finite p-groups with small authomorphism group (with J. González-Sánchez)

Forum of Mathematics, Sigma, Volume 3, 2015, e7 (autpgroups.pdf)

We show that there are non-abelian finite p-groups which the authomorphism group has smaller elements than the group itself. This gives an answer on a wel-known problem.

The
absolute
Galois group acts faithfully on regular dessins and on Beauville
surfaces (with **G.
Gónzalez-Diez**)

Proceedings of the
London Mathematical Society, 111 (2015), 775-796.
(short
version long
version)

A foundational result in Grothendieck's theory of dessins
d'enfants is the fact that the absolute Galois group G(Q)
of rational numbers acts faithfully on the set of all dessins.
However the question of whether this holds true when the action is
restricted to the set of the, more accessible, regular dessins
seems to be still an open question. In this paper we give an
affirmative answer to it. In fact we prove the strongest result that the
action is faithful on regular dessins of any fixed hyperbolic typy and
moreover G(Q) acts
faithfully on triangle (quasiplatonic) curves of any fixed hyperbolic
type. Furthermore, our methods allow us to prove two related
conjectures by Bauer, Catanese and Grunewald according to which 1) the
action of G(Q) on the set
of Beauville surfaces is faithful, and 2) for any element f
of G(Q) different from
the identity and the complex conjugation there is a Beaville surface S
such that S and its f-Galois
conjugate S^{f }have
non-isomorphic fundamental groups; the latter immediately implying that
the action of G(Q) on the
connected components of the moduli space of minimal surfaces of general
type is also faithful.

Property
(T) for groups graded by root systems **(**with Mikhail
Ershov and Martin
Kassabov)

Memoirs of the American Mathematical Society, 249 (2017), 1186. (rootsystems.pdf)

Abstract. We introduce and study the class of groups
graded by root sys-

tems. We prove that if X is an irreducible classical root system of
rank at least 2

and G is a group graded by X, then under certain natural conditions on
the

grading, the union of the root subgroups is a Kazhdan subset of G. As
the

main application of this result we prove that for any reduced
irreducible clas-

sical root system X of rank at least 2 and a finitely generated
commutative ring

R with 1, the Steinberg group St(X,R) and the elementary Chevalley
group

E(X,R) have property (T).

Groups graded by root systems and property (T) (with Mikhail Ershov, Martin Kassabov and Zezhou Zhang)

PNAS (2014); published ahead of print November 25, 2014, doi:10.1073/pnas.1321042111

**Normal
Subgroups of Profinite Groups of Non-negative Deficiency (with
Fritz Grunewald, Aline G.S. Pinto and Pavel A. Zalesski)**

* J. Pure Appl. Algebra
218 (2014), no. 5, 804–828.(normal.pdf)*

We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group G of non-negative deficiency gives rather strong consequences for the structure of G.

The representation zeta function of a FAb compact p-adic Lie group vanishes at -2 (with G. González-Sánchez and B. Klopsch)

Bull. Lond. Math. Soc. 46 (2014), no. 2, 239–244. (zeta-2.pdf)

Let G
be a compact p-adic Lie
group and suppose that G is
FAb,
i.e., every open subgroup G
has finite abelinization. The representation zeta function *ζ ^{G}(s)
=
∑r_{n}(G)n^{-s}
= ∑n_{i}^{-s}f_{i}(p^{-s}),
* where

Grafos, grupos y variedades: un punto de encuentro

La Gaceta de la RSME, Vol. 16 (2013), Núm. 4, Págs. 761–775 (expanders.pdf)

Groups of positive wighted deficiency and their applications (with Mikhail Ershov)J. Reine Angew. Math. 677 (2013), 71–134. (gosha.pdf )

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 Kourovka
notebook.

*Advances in
Mathematics, *227
(2011), 1129-1143 *(conjcl.pdf)*

**The rank
gradient from a combinatorial viewpoint (with Miklos Abert
and Nikolay Nikolov).**

Groups,
Geometry, and Dynamics*, *5
(2011), 213-230.* (combgr.pdf)*

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.

**Random generation of
finite and profinite groups and group enumeration (with Laci Pyber)**

*Annals of Matematics., 173
(2011), 769-814.
* (pfg.pdf)

**On Beauville surfaces
(with Y.
Fuertes and G.
Gónzalez-Diez)**

Groups,
Geometry, and Dynamics*, *5
(2011), 107-119.* (beauville.pdf)*

**
Property (T) for noncommutative universal lattices
(with Mikhail
Ershov)**

*Inventiones Mathematicae* 179
(2010), 303-347.* *(ELn.pdf)

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.

On p-groups having the minimal number of conjugacy classes of maximal size (with M.F. Newman and E.A. O'Brien)

* Israel
Journal of Mathematics
* 172
(2009), 119-123. (maxsize.pdf)

A long-standing question is the following: do there exist p-groups of odd order having precisely p − 1 conjugacy classes of the largest possible size? We exhibit a 3-group with this property.

**
Pro-p groups with few normal subgroups (with Y.
Barnea, N.
Gavioli, V. Monti, C.M. Scoppola)**

* Journal of
Algebra 321 (2009), 429-449.(fewnormal.pdf)*

Motivated by the study of pro-p groups of finite coclass, we consider the class of pro-p groups with few normal subgroups. This is not a well defined class and we offer several different definitions and study the connections between them. Furthermore, we propose a definition of periodicity for pro-p groups, thus, providing a general framework for some periodic patterns that have already been observed in the existing literature. We then focus on examples and show that strikingly all the interesting examples not only have few normal subgroups, but in addition have periodicity in the lattice of normal subgroups.

**On the verbal width of
finitely generated pro-p groups**

* Revista
Matemática
Iberoamericana
* **168** (2008), 393-412. (verbal.pdf)

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 is not contained in F''(F')^{p}. Also it is
shown that any word *w* has finite width in a
compact *p*-adic group.

**Omega
subgroups
of pro-p groups (with G.
Fernández-Alcober y J.
González-Sánchez)**

*Israel Journal of
Mathematics * * 166
(2008), 393-412*.

**Cohomological
properties
of the profinite completion of Bianchi groups (with F. Grunewald
and P. Zalesskii)**

* Duke
Mathematical Journal 144(2008), 53-72.
* (bianchi.pdf)

**On
linearity
of finitely generated R-analytic
groups.**

*Math.
Z. 253, No. 2, 333-345
(2006). *(linear.ps)

We prove that if *R* is a
commutative Noetherian local pro-*p *domain of
characteristic 0 then every finitely generated *R*-analytic
group is linear.

**Analytic
groups
over general pro-p domains (with ****B.
Klopsch****)**

*Journal London Math. Soc.
76(2007), 365-383. *(analytic.pdf)

**Zeta
function
of representations of compact p-adic analytic
groups. **

J. Amer. Math. Soc. 19 (2006) 91-118. (repr.ps)

We say that a profinite group* G* is **FAb** if
all open subgroups of *G* have finite abelinization. This
holds if and only if *r _{n}(G)=|{φ≤Irr(G)|φ(1)=n}|
*is finite for any

**On two
conditions on characters and conjugacy classes in finite soluble
groups****.**

* J.
Group
Theory 8 (2005), no.
3, 267--272. *(degree.ps)

We prove that there exists a
function* f(r)* such that the order of a soluble finite group G
is bounded by *f(r)* if one of the following conditions hold:

1. There exist at most r conjugacy classes in *G* of each
size.

2. There exist at most r irreducible characters in *G* of
each degree.

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

Proc.
Amer.
Math.
Soc. 133
(2005) 2817-2820. *
*(delta.ps)

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

* Chebyshevskii
Sb. 5 (2004), no.
1(9), 188--192. *(fake.pdf)

Let* J*
be a finite dimensional nilpotent algebra over a finite
field *F*. Then
the set *G=1+J* forms a finite group. The groups
constructed in this way is called **algebra
groups**. The group *G* acts by conjugation on *J*.
This induces an action of *G* on the dual space *J**.
The fake degree conjecture says that in
every algebra group *G=1+J* the character degrees
coincide, counting multiplicities, with the square roots of the
cardinals of the orbits of *J**. In this note we
construct a counterexample to this conjecture.

**The
number
of finite p-groups with bounded number of generators**

*Finite groups 2003, 209--217, Walter de Gruyter GmbH & Co. KG,
Berlin, 2004. *(def.dvi)

**In this note we
study the number of d-generated finite
p-groups.**

*J. of Algebra ***276 **
(2004), 193-209.*
*(potent.dvi)

Let *G* be a finite *p*-group
satisfying *[G,G]≤G ^{4}*

**On
the
number of conjugacy classes of finite p-groups. **

*Journal London Math. Soc* **68 **(2003),
699-711.(conj.dvi)

In this work we study the behaviour of the
number of conjugacy classes of finite p-groups using pro-p groups. We
introduce the conjugacy growth function r_{n}(*G*)=max
{
r(*G/N)|N◄G,|G:N|=n*}, where r(*G/N) *denotes
the number of conjugacy classes of *G/N*. We prove that
there are no infinite pro-p groups of linear conjugacy growth (i.e.
there is no *c* such that r_{n}(*G)≤c*log
*n* for all *n*>1) and we show that
many known pro-p groups *G* are of exponential
conjugacy growth (i.e. there exists a number *c=c(G)>0 *and
infinitely many open normal subgroups *N *of *G *such
that the number of conjugacy classes of *G/N* is greater than
*|G/N| ^{c} *).

**On
the
Growth of Noetherian Filtered Rings. (with ****D.
Pionkovskii****)**

* Communications in Algebra**
***31 **(2003),
505-512.(noet.dvi)

The goal of this note is to show that for
every Noetherian ring with a descending filtration its associated
graded ring grows subexponentially. The same is true for completed
group algebras of Noetherian pro-*p* groups and for group
algebras of Noetherian groups which are residually a finite *p*-group.
Also, we give a new simple proof of the Stephenson-Zhang theorem,
which asserts that Noetherian graded algebras grow subexponentially.

**On
the
number of conjugacy classes of finite p-groups of class 2.**

*preprint**
*(conjcl2.dvi)

In this work we study the behaviour of the
number of conjugacy classes of finite p-groups of class 2.

**Character
degrees
and nilpotence class of p-groups. (with **

Trans. Amer. Math. Soc. **354**
(2002), 3907-3925. (degree.pdf)

Let **U** be a finite set of
powers of *p* containing 1. It is known that for some choices
of **U**, if *P* is a finite *p*-group
whose set of character degrees is **U**, then the
nilpotence class of *P* is bounded by some integer that
depends on **U**, while for some other choices of **U**
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. The results obtained in these
papers made tempting to conjecture that a set **U** is
class bounding if and only if *p* doesnot belong to **U**.
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 **U** such that *p *doent
belong to **U**.

**On
linear
just infinite pro- p groups.**

*Journal of Algebra ***255**
(2002), 392-404* ** *(justinf.dvi)

In this work we prove that linear over
profinite rings just infinite pro-*p* groups and analytic just
infinite pro-*p* groups are linear over *Z** _{p}
*or

**Finite
groups
of bounded rank with an almost regular automorphisms. **

* **Israel**
Journal of Mathematics ***129**
(2002), 209-220* *(rank.pdf)

In this paper we prove that any finite
group of rank *r *with an automorphism, whose centralizer has
*m* points, has a characteristic soluble subgroup of *(m,r)*-bounded
index
and *r*-bounded derived length.

**A
connection between nilpotent groups and Lie rings. (with ****E.
I. Khukhro****)**

*Sibirsk. Mat. Zh. ***41**(2000),
994-1008
(nilp.dvi)

Let *G *be a nilpotent group of
class *c*. We use the Baker--Hausdorff formula to define the
structure of a Lie ring (** Z**-algebra)

**On
almost
regular automorphisms of finite ***p***-groups.**

*Advances in Mathematics*
**153**(2000), 391-402. (autom.dvi)

In this paper we prove that there are
functions *f(p,m,n)* and *h(m)* such that any finite *p*-group
with an automorphism of order *p ^{n}*, whose
centralizer has

**On
the
abundance of finite ***p***-groups.**

*Journal Group Theory ***3**(2000),
225-231.
(abun.dvi)

In this paper we prove that for given prime
*p* and non-negative integer *a, *there are only
finitely many *p*-groups of abundance *a.*

**On
the
use of the Lazard correspondence in the classification of ***p***-groups
of
maximal class.(with ****A.
Vera
Lopez****)
**

*Journal of Algebra ***228**(2000),
477-490.
(lazard.dvi)

Let *G* be a *p*-group of
maximal class of order *p ^{m}*,

**Modules
over
Crossed products.**

*Journal of Algebra ***215**(1999),
114-134.
(crprod.dvi)

J. T. Stafford proved that any left ideal
of the Weyl algebra *A _{n}(K)* over a field

*Fundamentalnaya i prikladnaya matematika*
**1**(1995), 813-816. (gauss.ps)

This paper continues a series of
investigations, devoted to generalized forms of Gauss lemma and
Eisenstein criterion.