Author

# Andrei Jaikin-Zapirain

Other affiliations: University of the Basque Country, Charles III University of Madrid, Autonomous University of Madrid

Bio: Andrei Jaikin-Zapirain is an academic researcher from Spanish National Research Council. The author has contributed to research in topics: Group (mathematics) & Normal subgroup. The author has an hindex of 14, co-authored 61 publications receiving 835 citations. Previous affiliations of Andrei Jaikin-Zapirain include University of the Basque Country & Charles III University of Madrid.

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##### Papers

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TL;DR: In this article, the authors studied the representation growth function of a profinite group with respect to the condition that all derived subgroups of the group G are open, where G is a finitely generated pro-p group and G has the property FAb (that is, H/H, H] is finite for every open subgroup H of G).

Abstract: Let G be a profinite group. We denote by rn(G) the number of isomorphism classes of irreducible n-dimensional complex continuous representations of G (so that the kernel is open in G). Following [20], we call rn(G) the representation growth function of G. If G is a finitely generated profinite group, then rn(G) < ∞ for every n if and only if G has the property FAb (that is, H/[H, H] is finite for every open subgroup H of G) [1, Proposition 2]. In the case when G is a finitely generated pro-p group, the property FAb is equivalent to the condition that all derived subgroups G are open. In this paper we shall investigate the function

112 citations

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TL;DR: In this article, a new spectral criterion for Kazhdan's property (T) was established, which is applicable to a large class of discrete groups defined by generators and relations.

Abstract: 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 EL
n
(R), where n≥3 and R is an arbitrary finitely generated associative ring. We also strengthen some of the results on property (T) for Kac-Moody groups from (Dymara and Januszkiewicz in Invent. Math. 150(3):579–627, 2002).

88 citations

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TL;DR: The concept of goodness was introduced by J.-P. Serre in his book on Galois cohomology as discussed by the authors, which relates the cohomological groups of a group to those of its profinite completion.

Abstract: The concept of cohomological goodness was introduced by J.-P. Serre in his book on Galois cohomology [31]. This property relates the cohomology groups of a group to those of its profinite completion. We develop properties of goodness and establish goodness for certain important groups. We prove, for example, that the Bianchi groups (i.e., the groups PSL(2,O), where O is the ring of integers in an imaginary quadratic number field) are good. As an application of our improved understanding of goodness, we are able to show that certain natural central extensions of Fuchsian groups are residually finite. This result contrasts with examples of P. Deligne [5], who shows that the analogous central extensions of Sp(4,Z) do not have this property

76 citations

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TL;DR: In this article, the minimal number of generators of finite index subgroups in residually finite groups is investigated and three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups.

Abstract: 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.

58 citations

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TL;DR: In this paper, the authors studied the action of the absolute Galois group Gal(Q/Q) on dessins d'enfants and Beauville surfaces and proved that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type.

Abstract: In this article we study the action of the absolute Galois group Gal(Q/Q) on dessins d’enfants and Beauville surfaces. A foundational result in Grothendieck’s theory of dessins d’enfants is the fact that the absolute Galois group Gal(Q/Q) 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 the first part of this paper we give an affirmative answer to it. In fact we prove the strongest result that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type. Beauville surfaces are an important kind of algebraic surfaces introduced by Catanese. They are rigid surfaces of general type closely related to dessins d’enfants. Here we prove that for any σ ∈ Gal(Q/Q) different from the identity and the complex conjugation there is a Beauville surface S such that S and its Galois conjugate S have non-isomorphic fundamental groups. This in turn easily implies that the action of Gal(Q/Q) on the set of Beauville surfaces is faithful. These results were conjectured by Bauer, Catanese and Grunewald, and immediately imply that Gal(Q/Q) acts faithfully on the connected components of the moduli space of surfaces of general type, a result due to the above mentioned authors.

55 citations

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582 citations

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01 Jan 1998

TL;DR: In particular, the set of all regular elements of a ring A such that a + I is a regular element of A/I is denoted by c(I).

Abstract: Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

369 citations

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TL;DR: In this paper, the authors summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

Abstract: We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

336 citations