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23
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 19 (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 14 (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
GolodShafarevich groups with property (T) and KacMoody groups, submitted
, 2006
"... Abstract. We construct GolodShafarevich groups with property (T) and thus provide counterexamples to a conjecture stated in a recent paper of Zelmanov [Ze2]. Explicit examples of such groups are given by lattices in certain topological KacMoody groups over finite fields. We provide several applic ..."
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Cited by 9 (1 self)
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Abstract. We construct GolodShafarevich groups with property (T) and thus provide counterexamples to a conjecture stated in a recent paper of Zelmanov [Ze2]. Explicit examples of such groups are given by lattices in certain topological KacMoody groups over finite fields. We provide several applications of this result including examples of residually finite torsion nonamenable groups. 1.
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 9 (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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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)
An infinite dimensional affine nil algebra with finite GelfandKirillov dimension∗
"... The famous 1960’s construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not nilporent, algebra which has polynomially bounded growth. ..."
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The famous 1960’s construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not nilporent, algebra which has polynomially bounded growth.
Burnside’s Problem, spanning trees and tilings
"... In this paper we study geometric versions of Burnside’s Problem and the von Neumann Conjecture. This is done by considering the notion of a translationlike action. Translationlike actions were introduced by Kevin Whyte as a geometric analogue of subgroup containment. Whyte proved a geometric vers ..."
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In this paper we study geometric versions of Burnside’s Problem and the von Neumann Conjecture. This is done by considering the notion of a translationlike action. Translationlike actions were introduced by Kevin Whyte as a geometric analogue of subgroup containment. Whyte proved a geometric version of the von Neumann Conjecture by showing that a finitely generated group is nonamenable if and only if it admits a translationlike action by any (equivalently every) nonabelian free group. We strengthen Whyte’s result by proving that this translationlike action can be chosen to be transitive when the acting free group is finitely generated. We furthermore prove that the geometric version of Burnside’s Problem holds true. That is, every finitely generated infinite group admits a translationlike action by Z. This answers a question posed by Whyte. In pursuit of these results we discover an interesting property of Cayley graphs: every finitely generated infinite group G has some locally finite Cayley graph having a regular spanning tree. This regular spanning tree can be chosen to have degree 2 (and hence be a biinfinite Hamiltonian path) if and only if G has finitely many ends, and it can be chosen to have any degree greater than 2 if and only if G is nonamenable. We use this last result to then study tilings of groups. We define a general notion of polytilings and extend the notion of MT groups and ccc groups to the setting of polytilings. We prove that every countable group is polyMT and every finitely generated group is polyccc. 20F65; 05C25, 05C63 1