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Peter Abramenko, Kenneth S Brown
June 18, 2007

### Abstract

We study two transitivity properties for group actions on buildings, called Weyl transitivity and strong transitivity. Following hints by Tits, we give examples involving anisotropic algebraic groups to show that strong transitivity is strictly stronger than Weyl transitivity. A surprising feature of the examples is that strong transitivity holds more often than expected.

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Gabriel Navarro
June 18, 2007

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Bálint Birszki
June 18, 2007

### Abstract

Sharp permutation groups are a generalization of sharply k -transitive permutation groups. In this article we show that, apart from sharply k -transitive groups, there are only a few primitive permutation groups which are sharp. In addition, we give a classification of solvable primitive sharp permutation groups.

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S Dolfi, M Herzog, G Kaplan, A Lev
June 18, 2007

### Abstract

Let G be a finite non-abelian group satisfying Φ( G ) = 1 and denote by U the nilpotent residual of G . In this paper, we prove that if G is of odd order then , and if G is of even order not divisible by a Mersenne or a Fermat prime then . These results are best possible and the assumption Φ( G ) = 1 cannot be omitted.

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Francesco Fumagalli
June 18, 2007

### Abstract

We study the commuting complex associated to the set of all non-trivial elements of a finite group. In particular we treat the case of metanilpotent groups, proving a wedgedecomposition formula for this simplicial complex and necessary and sufficient conditions for its contractibility.

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Norberto Gavioli, Valerio Monti, Carlo Maria Scoppola
June 18, 2007

### Abstract

A pro- p -group G is said to be normally constrained (or, equivalently, of obliquity zero ) if every open normal subgroup of G is trapped between two consecutive terms of the lower central series of G . In this paper infinite soluble normally constrained pro- p- groups, for an odd prime p , are shown to be 2-generated. A classification of such groups, up to the isomorphism type of their associated Lie algebra, is provided in the finite coclass case, for p > 3. Moreover, we give an example of an infinite soluble normally constrained pro- p- group whose lattice of open normal subgroups is isomorphic to that of the Nottingham group. Some general results on the structure of soluble just infinite pro- p- groups are proved on the way.

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Andre Nies
June 18, 2007

### Abstract

An infinite f.g. group G is quasi-finitely axiomatizable (QFA) if there is a first-order sentence ϕ such that G ⊨ ϕ, and if H is a f.g. group such that H ⊨ ϕ, then G ≅ H . The first result is that all Baumslag–Solitar groups of the form 〈 a, d | d -1 ad = a m 〉 are QFA. A f.g. group G is a prime model if and only if there is a tuple g 1 , … , g n generating G whose orbit (under the automorphisms of G ) is definable by a first-order formula. The second result is that there are continuum many non-isomorphic f.g. groups that are prime models. In particular, not all are QFA.

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Diego Rattaggi
June 18, 2007

### Abstract

We construct a finitely presented torsion-free simple group Σ 0 , acting cocompactly on a product of two regular trees. An infinite family of such groups was introduced by Burger and Mozes [M. Burger and S. Mozes. Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris Sér. I Math . 324 (1997), 747–752.], [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math . 92 (2001), 151–194.]. We refine their methods and construct Σ 0 as an index 4 subgroup of a group presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math . 92 (2001), 151–194., Theorem 6.4] needs more than 18000 relations, and the smallest simple group constructed in [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math . 92 (2001), 151–194., §6.5] needs even more than 360000 relations in any finite presentation.

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Adam Piggott
June 18, 2007

### Abstract

For an integer n at least two and a positive integer m , let C( n,m ) denote the group of Andrews–Curtis transformations of rank ( n,m ) and let F denote the free group of rank n + m . A subgroup AC( n,m ) of Aut(F) is defined, and an anti-isomorphism AC( n,m ) to C( n,m ) is described. We solve the generalized word problem for AC( n,m ) in Aut(F) and discuss an associated reformulation of the Andrews–Curtis conjecture.

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P. C Wong, K. B Wong
June 18, 2007

### Abstract

We prove that the outer automorphism groups of tree products of finitely many polycyclic-by-finite groups amalgamating central edge groups are residually finite. As a consequence, the outer automorphism groups of tree products of finitely many abelian groups are residually finite.

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Bartosz Putrycz
June 18, 2007

### Abstract

A generalized Hantzsche–Wendt (GHW) group is by definition the fundamental group of a flat n -manifold with holonomy group ℤ 2 n -1 , and a Hantzsche–Wendt (HW) group is a GHW group corresponding to an orientable manifold (with n odd). We prove that for any n -dimensional HW group with n > 3, the commutator subgroup and translation subgroup are equal, and hence the abelianization of the group is ℤ 2 n -1 . We also give examples of GHW groups with the same property for all n > 4. These groups are all examples of torsion-free metabelian groups with abelianizations ℤ 2 k for some k > 3.