Results 1  10
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14
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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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.
Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Cited by 17 (1 self)
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Some cases of the FontaineMazur conjecture
 J. Number Theory
, 1992
"... Abstract. We prove more special cases of the FontaineMazur conjecture regarding padic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it. 0. Introduction. Answering a longstanding question of Furtwängler, mentioned as early as 1926 [10], ..."
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Cited by 12 (3 self)
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Abstract. We prove more special cases of the FontaineMazur conjecture regarding padic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it. 0. Introduction. Answering a longstanding question of Furtwängler, mentioned as early as 1926 [10], Golod and Shafarevich showed in 1964 [8] that there exists a number field with an infinite, everywhere unramified prop extension. In fact it is easy to obtain many examples [8], [13], [23]. Very little is known, however, regarding the structure
Noncommutative complete intersections and matrix integrals
, 2007
"... We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Repn A, of ndimensional representations of A, is a complete inters ..."
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Cited by 8 (4 self)
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We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Repn A, of ndimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is another interesting class of associative algebras called noncommutative complete intersections (NCCI). We prove that any graded RCI algebra is NCCI. We also obtain explicit formulas for the Hilbert series of each nonvanishing cyclic and Hochschild homology group of an RCI algebra. The proof involves a noncommutative cyclic Koszul complex, K cyc � A, and a matrix integral similar to the one arising in quiver gauge theory.
Infinite Global Fields and the Generalized Brauer–Siegel Theorem
 Moscow Math. J
"... To our teacher Yu.I.Manin on the occasion of his 65th birthday Abstract. The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of Q or of Fr(t). We produce a series of invariants of such fields, and we introduce and s ..."
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Cited by 6 (2 self)
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To our teacher Yu.I.Manin on the occasion of his 65th birthday Abstract. The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of Q or of Fr(t). We produce a series of invariants of such fields, and we introduce and study a kind of zetafunction for them. Second, for sequences of number fields with growing discriminant we prove generalizations of the Odlyzko–Serre bounds and of the Brauer–Siegel theorem, taking into account nonarchimedean places. This leads to asymptotic bounds on the ratio log hR / log √ D  valid without the standard assumption n / log √ D  → 0, thus including, in particular, the case of unramified towers. Then we produce examples of class field towers, showing that this assumption is indeed necessary for the Brauer–Siegel theorem to hold. As an easy consequence we ameliorate on existing bounds for regulators. 2000 Math. Subj. Class. 11G20, 11R37, 11R42, 14G05, 14G15, 14H05 Key words and phrases. Global field, number field, curve over a finite field, class number, regulator, discriminant bound, explicit formulae, infinite global field, Brauer–Siegel theorem 1
Groups of PrimePower Order With a Small Number of Relations
, 1995
"... this paper are: 142 Newman, Sauerbier, Wisliceny For every prime p there is a proppresentation with 5 generators and 7 relations which defines a finite propgroup with generator number 5. (Theorem 4.2 and Remark 4.3) ..."
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this paper are: 142 Newman, Sauerbier, Wisliceny For every prime p there is a proppresentation with 5 generators and 7 relations which defines a finite propgroup with generator number 5. (Theorem 4.2 and Remark 4.3)
Counting manifolds and class field towers
"... Abstract. In [BGLM] and [GLNP] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x (γ(H)+o(1)) log x/log log x where γ(H) is an explicit constant computable from the (absolute) root ..."
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Abstract. In [BGLM] and [GLNP] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x (γ(H)+o(1)) log x/log log x where γ(H) is an explicit constant computable from the (absolute) root system of H. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate x c log x. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers. 1.
COUNTING THE PERIODIC GROUPS GENERATED BY TWO FINITE GROUPS
"... exponent p 2 for each sufficiently large prime p. The proof uses wreath products and embedding methods and relies on the existence of the infinite pgroups of Novikov and Adjan [7]. Phillips [8] had previously proved the existence of 2 N ° nonisomorphic twogenerator ..."
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exponent p 2 for each sufficiently large prime p. The proof uses wreath products and embedding methods and relies on the existence of the infinite pgroups of Novikov and Adjan [7]. Phillips [8] had previously proved the existence of 2 N ° nonisomorphic twogenerator
An infinite dimensional affine nil algebra with finite
, 2005
"... We construct a nil algebra over a countable field which has finite but nonzero GelfandKirillov dimension. 2000 Mathematics subject classification: 16N, 16P90. ..."
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We construct a nil algebra over a countable field which has finite but nonzero GelfandKirillov dimension. 2000 Mathematics subject classification: 16N, 16P90.
TAME PROp GALOIS GROUPS: A SURVEY OF RECENT WORK
, 2004
"... Fix a prime p, a number field K, and a finite set S of places of K none of which has residue characteristic p. Fix an algebraic closure K of K and let KS be the maximal pextension of K inside K which is unramified outside S; it is the compositum of all finite ppower degree extensions of K unramifi ..."
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Fix a prime p, a number field K, and a finite set S of places of K none of which has residue characteristic p. Fix an algebraic closure K of K and let KS be the maximal pextension of K inside K which is unramified outside S; it is the compositum of all finite ppower degree extensions of K unramified outside S. We assume that real places of K not contained in S do not complexify in the extension KS/K. Put GK,S = Gal(KS/K) for its (prop) Galois group. Very little is known about this “tame arithmetic fundamental group. ” Before Shafarevich’s pioneering work [Sh], a few examples where it was possible to determine GK,S explicitly (and show that it was finite), were known, and it was in fact generally believed that all such GK,S are finite. That this is not so was first demonstrated in [GS] by Golod and Shafarevich. As was noted by Artin and Shafarevich, the mere existence of infinite GK,S (with S finite) has an arithmetic application to the estimation of discriminants because the discriminants of successive fields in a tamely and finitely ramified tower grow as slowly as possible. For a more detailed discussion of this topic (and the analogy with curves over finite fields with many rational points) see, for example, [HM1] and the references therein. Infinite GK,S satisfy a number of interesting grouptheoretic properties (stemming from