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Daniel Farley
March 15, 2011

### Abstract

Each of Thompson's groups F, T , and V has infinitely many ends relative to the groups F [0, 1/2] , T [0, 1/2] , and V [0, 1/2) (respectively). We can therefore simplify the proof, due to Napier and Ramachandran, that F, T , and V are not Kähler groups. Thompson's groups T and V have Serre's property FA. The original proof of this fact was due to Ken Brown.

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Benjamin Fine, Aila Rosenberger, Gerhard Rosenberger
August 29, 2011

### Abstract

Lyndon in [Combinatorial Group Theory, Springer-Verlag, 1977] proved that in a free group F each solution set { x, y, z } of the quadratic equation x 2 y 2 z 2 = 1 generates a cyclic group. This theorem launched the whole theory of equations over free groups which led eventually to the solution of the Tarski conjectures. Since Lyndon's theorem can be phrased as a universal sentence, it is true in all fully residually free groups, if we replace cyclic by abelian. In the present paper we consider amalgams of groups which satisfy Lyndon's theorem. We prove that this property and several other related properties involving quadratic and quadratic-type equations are preserved under free products with malnormal amalgamated subgroups. The situation in HNN groups is more complicated.

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Ilya Kapovich, Martin Lustig
August 29, 2011

### Abstract

We prove that if T is an ℝ-tree with a minimal free isometric action of F N , then the Out( F N )-stabilizer of the projective class [ T ] is virtually cyclic. For the special case where T = T + ( ϕ ) is the forward limit tree of an atoroidal iwip element ϕ ∈ Out( F N ) this is a consequence of the results of Bestvina, Feighn and Handel [Geom. Funct. Anal. 7: 215–244, 1997], via very different methods. We also derive a new proof of the Tits alternative for subgroups of Out( F N ) containing an iwip (not necessarily atoroidal): we prove that every such subgroup G ⩽ Out( F N ) is either virtually cyclic or contains a free subgroup of rank two. The general case of the Tits alternative for subgroups of Out( F N ) is due to Bestvina, Feighn and Handel.

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Wenbin Guo, Alexander N. Skiba
August 29, 2011

### Abstract

A chief factor H / K of a group G is called ℱ-central if [ H / K ]( G / C G ( H / K )) ∈ ℱ. The product of all such normal subgroups of G , whose G -chief factors are ℱ-central in G , is called the ℱ-hypercentre of G and denoted by . The finite groups G with factorizations G = AB , where for some class of groups ℱ, are studied. Some known results about factorizations of groups are generalized.

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Lars Pforte
August 29, 2011

### Abstract

For any group G the involutions ℐ in G form a G -set under conjugation. The corresponding kG -permutation module k ℐ is known as the involution module of G , with k an algebraically closed field of characteristic two. In this paper we discuss the involution module of the special linear group SL 2 (2 ƒ ). We describe the vertices and the Loewy and Socle Series of all its components.

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Matjaž Konvalinka, Götz Pfeiffer, Claas E. Röver
August 29, 2011

### Abstract

The normalizer N W ( W J ) of a standard parabolic subgroup W J of a finite Coxeter group W splits over the parabolic subgroup with complement N J consisting of certain minimal length coset representatives of W J in W . In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type D n ) the centralizer C W ( w ) of an element w ∈ W is in a similar way a semidirect product of the centralizer of w in a suitable small parabolic subgroup W J with complement isomorphic to the normalizer complement N J . Then we use this result to give a new short proof of Solomon's Character Formula and discuss its connection to MacMahon's master theorem.

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Siyka Andreeva, Roland Schmidt, Imke Toborg
August 29, 2011

### Abstract

If L is a lattice, a group is called L -free if its subgroup lattice has no sublattice isomorphic to L . It is easy to see that for every sublattice L of L 10 , the subgroup lattice of the dihedral group of order 8, the finite L -free groups form a lattice-defined class of groups with modular Sylow subgroups. In this paper we determine the structure of these groups for the two non-modular 8-element sublattices of L 10 .

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Ahmet Arıkan, Howard Smith
September 1, 2011

### Abstract

We consider minimal non-(finite exponent) groups and give certain characterizations of these groups. We also prove certain results relevant to Fitting p -groups with all proper subgroups solvable.

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Tom Edgar
August 29, 2011

### Abstract

We provide a case-free characterization of hyperbolic Coxeter systems depending only on the Coxeter graph. Consequently, we uncover a case-free proof of the fact that every infinite, non-affine Coxeter system contains a standard parabolic subsystem that is a hyperbolic Coxeter system.

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Antonio Beltrán, Mariá José Felipe
August 29, 2011

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Ticianne P. Bueno, Noraí R. Rocco
March 15, 2011

### Abstract

We study the non-abelian tensor square modulo q of a group, where q is a non-negative integer, via an operator ν q in the class of groups. Structural properties and finiteness conditions of ν q ( G ) are investigated. We compute the non-abelian tensor square modulo q of cyclic groups and develop a theory for computing ν q ( G ) and some of its relevant sections for polycyclic groups G . This extends the existing theory from the case q = 0 to all non-negative integers q . Additionally, a table of examples is produced with the help of the GAP system.