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99
Closed subgroups in proV topologies and the extension problem for inverse automata
 INT. J. ALGEBRA COMPUT
, 1999
"... We relate the problem of computing the closure of a finitely generated subgroup of the free group in the proV topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial onetoone maps on a finite set, can they ..."
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Cited by 41 (7 self)
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We relate the problem of computing the closure of a finitely generated subgroup of the free group in the proV topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial onetoone maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extensionclosed. Turning to practical computations, we modify Ribes and Zalesski i's algorithm to compute the prop closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pronilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups.
Finite covers of random 3manifolds
, 2005
"... A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed ..."
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Cited by 39 (1 self)
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A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface Σ on
A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory
 J. Reine angew. Math
"... Abstract. In this paper and a forthcoming joint one with Y. Hachimori [15] we study Iwasawa modules over an infinite Galois extension k ∞ of a number field k whose Galois group G = G(k∞/k) is isomorphic to the semidirect product of two copies of the padic integers Zp. After first analyzing some gen ..."
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Cited by 32 (5 self)
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Abstract. In this paper and a forthcoming joint one with Y. Hachimori [15] we study Iwasawa modules over an infinite Galois extension k ∞ of a number field k whose Galois group G = G(k∞/k) is isomorphic to the semidirect product of two copies of the padic integers Zp. After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group Xnr of the pHilbert class field over k∞. The main tool we use is a version of the Weierstrass preparation theorem, which we prove for certain skew power series with coefficients in a not necessarily commutative local ring. One striking result in our work is the discovery of the abundance of faithful torsion Λ(G)modules, i.e. nontrivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra Fp[G] with coefficients in the finite field Fp is a unique factorization domain in the sense of Chatters [8]. 1.
Topological Tits alternative
 the Annals of Math
, 2004
"... Abstract. Let k be a local field, and Γ ≤ GLn(k) a linear group over k. We prove that either Γ contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups. 1. ..."
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Cited by 32 (12 self)
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Abstract. Let k be a local field, and Γ ≤ GLn(k) a linear group over k. We prove that either Γ contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups. 1.
On the structure theory of the Iwasawa algebra of a padic Lie group
 J. Eur. Math. Soc
"... Abstract. This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, Λ of a padic analytic group G. For G without any ptorsion element we prove that Λ is an Auslander regular ring. This ..."
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Cited by 30 (4 self)
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Abstract. This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, Λ of a padic analytic group G. For G without any ptorsion element we prove that Λ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudonull Λmodule. This is classical when G = Zk p for some integer k ≥ 1, but was previously unknown in the noncommutative case. Then the category of Λmodules up to pseudoisomorphisms is studied and we obtain a weak structure theorem for the Zptorsion part of a finitely generated Λmodule. We also prove a local duality theorem and a version of AuslanderBuchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere.
Completely Faithful Selmer Groups over Kummer Extensions
 DOCUMENTA MATH.
, 2003
"... In this paper we study the Selmer groups of elliptic curves over Galois extensions of number fields whose Galois group G ∼ = Zp ⋊ Zp is isomorphic to the semidirect product of two copies of the padic numbers Zp. In particular, we give examples where its Pontryagin dual is a faithful torsion modu ..."
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Cited by 26 (6 self)
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In this paper we study the Selmer groups of elliptic curves over Galois extensions of number fields whose Galois group G ∼ = Zp ⋊ Zp is isomorphic to the semidirect product of two copies of the padic numbers Zp. In particular, we give examples where its Pontryagin dual is a faithful torsion module under the Iwasawa algebra of G. Then we calculate its Euler characteristic and give a criterion for the Selmer group being trivial. Furthermore, we describe a new asymptotic bound of the rank of the MordellWeil group in these towers of number fields.
Representation Growth for Linear Groups
"... Abstract. Let Γ be a group and rn(Γ) the number of its ndimensional irreducible complex representations. We define and study the associated representation zeta function ZΓ(s) = ∞∑ rn(Γ)n−s. When Γ is an arithmetic n=1 group satisfying the congruence subgroup property then ZΓ(s) has an “Euler facto ..."
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Cited by 23 (2 self)
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Abstract. Let Γ be a group and rn(Γ) the number of its ndimensional irreducible complex representations. We define and study the associated representation zeta function ZΓ(s) = ∞∑ rn(Γ)n−s. When Γ is an arithmetic n=1 group satisfying the congruence subgroup property then ZΓ(s) has an “Euler factorization”. The “factor at infinity ” is sometimes called the “Witten zeta function ” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups U of the associated simple group G over the associated local field K. Here we show a surprising dichotomy: if G(K) is compact (i.e. G anisotropic over K) the abscissa of convergence goes to 0 when dim G goes to infinity, but for isotropic groups it is bounded away from 0. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa. 1.
Lie methods in growth of groups and groups of finite width
, 2000
"... In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an Nseries of subgroups. The asymptotics of the Poincaré series of this algebra give estimates on the growth of the group G. This establishes the existence of a gap between polynomial ..."
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Cited by 22 (15 self)
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In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an Nseries of subgroups. The asymptotics of the Poincaré series of this algebra give estimates on the growth of the group G. This establishes the existence of a gap between polynomial growth and growth of type e √ n in the class of residually–p groups, and gives examples of finitely generated p–groups of uniformly exponential growth. In the second part, we produce two examples of groups of finite width and describe their Lie algebras, introducing a notion of Cayley graph for graded Lie algebras. We compute explicitly their lower central and dimensional series, and outline a general method applicable to some other groups from the class of branch groups. These examples produce counterexamples to a conjecture on the structure of justinfinite groups of finite width.
The homotopy fixed point spectra of profinite Galois extensions
"... Let E be a klocal profinite GGalois extension of an E∞ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete Gspectrum. Also, we prove that if E is a profaithful klocal profinite extension which satisfies certain extra conditions, then the forward dir ..."
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Cited by 22 (15 self)
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Let E be a klocal profinite GGalois extension of an E∞ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete Gspectrum. Also, we prove that if E is a profaithful klocal profinite extension which satisfies certain extra conditions, then the forward direction of Rognes’s Galois correspondence extends to the profinite setting. We show the function spectrum FA((EhH)k, (EhK)k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]]) hK)k where H and K are closed subgroups of G. Applications to Morava Etheory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points.
Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre
 , TO APPEAR IN VOLUME V OF MILNOR’S COLLECTED WORKS
, 2008
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