Please follow the IMU's recommendation and make ALL your Math papers available electronically.

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

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

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

Approximation by subgroups
of finite index and the Hanna Neumann conjecture

The base change in the Atiyah and the Lück approximation conjectures

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

The universality of Hughes-free division rings

An explicit construction of the universal division ring of fractions of E<<x1,...xd>>

Free Q-groups are residually torsion-free nilpotent.

**All
publications (by subjects)**

The universality of Hughes-free division rings

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

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

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

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

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

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

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

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

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

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

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

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

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

Approximation by subgroups of finite index and the Hanna Neumann conjecture

Recognition of being fibered for compact 3-manifolds

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

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

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

**Appendix
to
Ershov's paper KAZHDAN QUOTIENTS OF
GOLOD-SHAFAREVICH GROUPS**

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

The base change in the Atiyah and the Lück approximation conjectures

L2-Betti numbers and their analogues in positive characteristic

** Property (T) for noncommutative
universal lattices
(with
M. Ershov)**

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

Groups
of positive weightd deficiency and their applications** (with
M.
Ershov)**

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

Approximation by subgroups of finite index and the Hanna Neumann conjecture

The base change in the Atiyah and the Lück approximation conjectures

L2-Betti numbers and their analogues in positive characteristic

Free Q-groups are residually torsion-free nilpotent.

**Groups**** IN aLGEBRAIC
GEOMETRY **

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

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

**Divulgacion**

Grafos, grupos y variedades: un punto de encuentro

Free Q-groups are residually torsion-free nilpotent.

preprint (baumslag.pdf)

We show that a free Q-group is residually torsion-free nilpotent. This solves a 50 years old problem proposed by G. Baumslag.

back

An explicit construction of the universal division ring of fractions of E<<x1,...xd>>.

preprint (sylvester.pdf)

We give a sufficient and necessary condition for a Sylvester matrix rank function on a ring to be equal to its inner rank. We apply this criterion in different contexts. The main application is an explicit contruction of a universal division ring of fractions of E<<x1,...,xd>>.

The universality of Hughes-free division rings

preprint (universal.pdf)

Let E be a division ring and G a locally indicable group. We prove that the Hughes-free division E*G-ring is universal in the sense of Cohn if G is either amenable or residually torsion-free nilpotent.

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)

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.