J. T. Stafford proved that any left ideal of the Weyl algebra A_n(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.

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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, p an odd prime and m>3. In this work we reduce the construction of this group to the consideration of certain elements of Hom_S(R/a^m-2/\R/a^m-2, R/a^m-4), where R=Z[x]/(1+...+x^p-1), a=(x-1) and S=Z[ x]/(x^p-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.

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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.

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

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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.

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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.

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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 F_p[[t]].

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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.

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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.

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

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

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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)≤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 ).

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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.

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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.

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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.

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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]≤G⁴for p=2 and γ_p-1(G)≤ G^p for p>2 . The main goal of this paper is to show that any normal subgroup of G lying in G² is power abelian.

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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.

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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 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,...., n_k and functions f₁(p^-s),..., f_k(p^-s) rational in p^-ssuch that ζ^G(s) = ∑r_n(G)n^-s = ∑n_i^-sf_i(p^-s) .

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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.

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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.

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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.

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

Annals of Math., to appear (pfg.pdf)

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

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

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

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

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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.

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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 with nite 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.

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

preprint (beauville.pdf)

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

Inventiones Mathematicae, to appear (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.

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

preprint (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.

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

preprint (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.

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On the number of conjugacy classes of finite nilpotent groups

preprint (conjcl.pdf)

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

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