The complete list of publications before 2020 with abstracts in reverse chronological order.

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.


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)


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.


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.


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:


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 such that S and its f-Galois conjugate  Sf 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).

For a short exposition of this paper see:

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 ζG(s) = rn(G)n-s = ni-sfi(p-s),  where  rn(G)=|{φIrr(G)|φ(1)=n}|, encodes the distribution of continuous  irreducible complex characters of G.  For p>2 it is known that ζG(s) defines a meromorphic function on C. Wedderburn's structure theorem for semisimple algebras implies that ζG(-2)=|G| for finite G.  We complement this classic result by proving that ζG(-2) = 0 for infinite G assuming p>2.


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.


On the number of conjugacy classes of finite nilpotent groups

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. (omega.pdf)


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).  (

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. (

We  say that a profinite group G is  FAb if all  open subgroups of G have finite abelinization. This holds  if and only if  rn(G)=|{φIrr(G)|φ(1)=n}| is finite for any n1. Let  G  be a FAb compact  p-adic analytic group and suppose that  p>2  or  p=2  and  G  is uniform. In this note we  prove that   there exist natural numbers n1,...., nk and functions  f1(p-s),..., fk(p-s) rational in p-s such that  ζG(s) = rn(G)n-s = ni-sfi(p-s) .


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

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

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 (

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.


On the fake degree conjecture

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.


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

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

Let G be a finite p-group satisfying [G,G]≤G4 for  p=2  and γp-1(G)≤ Gp for  p>2 . The main goal of this paper is to show that   any normal subgroup of  G  lying in G2  is power abelian.


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 rn(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 rn(G)≤clog 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 A. Moretó)

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 Zp or Fp[[t]].


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) M on the subgroup Gn, for some n=n(c) depending only on c, in such a way that many important parameters of M, like the nilpotency class and the derived length, are equal to those of Gnas a group. As an application we refine reductions of theorems about "almost regular" p-automorphisms of finite p-groups  to corresponding theorems on Lie rings. In particular, we prove that the m-bounded function in Medvedev's theorem on p-groups with an automorphism of order p can be chosen to be exactly the same as in his theorem on Lie rings. Besides, we show that Higman's and Kreknin's functions that appear in results on fixed-point-free automorphisms of Lie algebras are the best possible bounds (if required to depend only on the order of the automorphism) for the nilpotency class and the derived length respectively of a subgroup of bounded index in theorems on p-automorphisms of finite p-groups.


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 pn, whose centralizer has pm points, has a subgroup of derived length at most h(m) and index at most f(p,m,n).


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 pm, p an odd prime and m>3. In this work we reduce the construction of this group to the consideration of certain elements of HomS(R/am-2/\R/am-2, R/am-4), where R=Z[x]/(1+...+xp-1), a=(x-1) and S=Z[ x]/(xp-1). As an application of this result we prove that the structure of G is determined by the (p-3)/2 commutators and three invariants.


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 An(K) over a field K of characteristic zero can be generated by two elements. In general, there is the problem of determining whether any left ideal of a Noetherian simple domain can be generated by two elements. In this work we show that this property holds for some crossed products of simple ring with a supersolvable group and also for the tensor product of generalized Weyl algebras.


Weak graded analogues of Gauss lemma and Eisenstein criterion.

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

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


Departamento de Matemáticas, Facultad de Ciencias, 
Universidad Autónoma de Madrid, 
Cantoblanco Ciudad Universitaria 
28049 Madrid, Spain.


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