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
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 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).
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)=|{φ≤Irr(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.
backGrafos, 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. (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). (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 rn(G)=|{φ≤Irr(G)|φ(1)=n}| is finite for any n≥1. 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. (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.
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.
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
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. (gauss.ps)
This paper continues a series of
investigations, devoted to generalized forms of Gauss lemma and
Eisenstein criterion.
