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Moshe Jarden
May 16, 2006

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Abderezak Ould Houcine
May 16, 2006

### Abstract

We prove that a δ-hyperbolic group for δ < ½ is a free product F * G 1 * … * G n where F is a free group of finite rank and each G i is a finite group.

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Gábor Elek, Endre Szabó
May 16, 2006

### Abstract

Answering some questions of Weiss in [B. Weiss. Sofic groups and dynamical systems. (Ergodic theory and harmonic analysis, Mumbai, 1999.) Sankhya Ser. A. 62 (2000), 350-359.], we prove that a free product of sofic groups is sofic and that amenable extensions of sofic groups are sofic. We also give an example of a finitely generated sofic group that is not residually amenable.

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James Howie, Natalia Kopteva
May 16, 2006

### Abstract

A generalized tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalized triangle groups and whose edge groups are finite cyclic. We prove an improved spelling theorem for generalized triangle groups which enables us to compute the precise Gersten–Stallings angles of this triangle of groups, and hence obtain a classification of generalized tetrahedron groups according to the curvature properties of the triangle. We also prove that the colimit of a negatively curved triangle of groups contains a non-abelian free subgroup. Finally, we apply these results to prove the Tits alternative for all generalized tetrahedron groups where the triangle is non-spherical: with three abelian-by-finite exceptions, every such group contains a non-abelian free subgroup.

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Gilbert Baumslag, Sean Cleary
May 16, 2006

### Abstract

Parafree groups are groups that are residually nilpotent and have the property that their quotients by the terms of the lower central series are isomorphic to the corresponding quotients of a free group. We introduce three new families of non-free parafree groups and discuss limitations to a natural procedure for distinguishing these groups from each other.

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Eloisa Detomi, Andrea Lucchini
May 16, 2006

### Abstract

We discuss whether finiteness properties of a profinite group G can be deduced from the probabilistic zeta function P G ( s ). In particular we prove that in the prosoluble case, if P G ( s ) is rational then G /Frat( G ) is finite.

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Pablo Spiga
May 16, 2006

### Abstract

A partition for the elements of prime-power order in a finite group G is a family of subgroups with the property that every non-identity element of prime-power order lies in exactly one subgroup of the family. The main result of this paper is a classification of the finite simple groups which have such a partition. We also establish a connection between this concept and the class of permutation groups all of whose elements of prime-power order have the same number of fixed points.

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Giovanni Cutolo
May 16, 2006

### Abstract

This paper deals with an old problem: are there non-trivial finite p -groups which are isomorphic to their full automorphism group, besides the dihedral group of order 8? The answer (in the negative) is obtained in some special cases, including groups of class 2, powerful groups, groups with centre of prime order or an abelian subgroup of prime index, class-3 groups with cyclic centre, groups with coclass at most 3 and others.

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Jianbei An, Gerhard Hiss
May 16, 2006

### Abstract

We determine the restriction of the Steinberg character of a finite symplectic group of odd characteristic to its maximal parabolic subgroup stabilizing a line. We relate this restriction to the tensor product of a Weil character with the Steinberg character. As an application we prove that Donovan's conjecture has a positive answer for unipotent l -blocks of the six-dimensional symplectic groups of odd characteristic when l > 3.

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Angel Carocca, Víctor González-Aguilera, Rubí E Rodríguez
May 16, 2006

### Abstract

Let G be a finite group. For each integral representation ρ of G we consider ρ-decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties ( X, H ) with -action, of dimension equal to the degree of ρ, which admit a decomposition of the lattice for X into two G -invariant sublattices isotropic with respect to ℑ H , with one of the sublattices ℤ G -isomorphic to ρ. We give a construction for ρ-decomposable principally polarized abelian varieties, and show that each of them is isomorphic to a product of elliptic curves. Conversely, if ρ is absolutely irreducible, we show that each ρ-decomposable p.p.a.v. is (isomorphic to) one of those constructed above, thereby characterizing them. In the case of irreducible, reduced root systems, we consider the natural representation of its associated Weyl group, apply the preceding general construction, and characterize completely the associated families of principally polarized abelian varieties, which correspond to modular curves.