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Abderezak Ould Houcine, Françoise Point
July 1, 2013

### Abstract

Abstract. The famous Tits alternative states that a linear group either contains a non-abelian free group or is soluble-by-(locally finite). In this paper we study similar alternatives in pseudofinite groups. We show, for instance, that an ℵ 0 -saturated pseudofinite group either contains a subsemigroup of rank 2 or is nilpotent-by-(uniformly locally finite). We call a class of finite groups G weakly of bounded rank if the radical has a bounded Prüfer rank and the index of the socle of is bounded. We show that an ℵ 0 -saturated pseudo-(finite weakly of bounded rank) group either contains a non-abelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between these kind of alternatives and amenability.

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Yiftach Barnea, Jan-Christoph Schlage-Puchta
July 1, 2013

### Abstract

Abstract. Recently, Schlage-Puchta proved super-multiplicity of p -deficiency for normal subgroups of p -power index. We extend this result to all normal subgroups of finite index. We then use the methods of the proof to show that some groups with non-positive p -deficiency have virtually positive p -deficiency. We also compute the p -deficiency in some cases such as Fuchsian groups and study related invariants: the lower and upper absolute p -homology gradients and the p -Euler characteristic.

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Nathaniel Pappas
July 1, 2013

### Abstract

Abstract. For any prime number p and any positive real number we construct a finitely generated group Γ with p -gradient equal to . This construction is used to show that there exist uncountably many pairwise non-commensurable groups that are finitely generated, infinite, torsion, non-amenable, and residually- p .

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Colin D. Reid
July 1, 2013

### Abstract

Abstract. We define a local Sylow subgroup of a totally disconnected, locally compact (t.d.l.c.) group G to be a maximal pro- p subgroup of an open compact subgroup of G . We use these subgroups to define the p -localization of G , a locally virtually pro- p group which maps continuously and injectively to G with dense image, and describe the relationship between the scale and modular functions of G and those of its p -localization. In the case of locally virtually prosoluble groups, we consider all primes simultaneously using local Sylow bases.

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Stefaan Delcroix
July 1, 2013

### Abstract

Abstract. In this paper, we prove the following characterization of LFS-groups of p -type: Let G be a locally finite group and be a local system for G such that and for all : (i) is a local system for G . (ii) there exists such that (a) is block-diagonal for S for , (b) for and , (c) for , where is the number of composition factors C for on with . Then is an LFS-group of p -type and . We use this theorem to construct a general family of LFS-groups of p -type.

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Emanuele Rodaro, Pedro V. Silva, Mihalis Sykiotis
July 1, 2013

### Abstract

Abstract. It is shown, for a given graph group G , that the fixed point subgroup is finitely generated for every endomorphism of G if and only if G is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms if the dependence graph of G is a transitive forest.

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Zeinab Akhlaghi, Antonio Beltrán, María José Felipe
July 1, 2013

### Abstract

Abstract. Let G be a finite group and N be a normal subgroup of G and suppose that the p -regular elements of N have exactly two G -conjugacy class sizes. It is shown that N is solvable and that if H is a p -complement of N , then either H is abelian or H is the product of an r -group for some prime and a central subgroup of G .

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John J. Ballantyne
July 1, 2013

### Abstract

Abstract. Given a finite group G and G -conjugacy class of involutions X , the local fusion graph has X as its vertex set, with joined by an edge if, and only if, and the product has odd order. In this note we investigate such graphs when G is a finite Coxeter group, addressing questions of connectedness and diameter. In particular, our results show that local fusion graphs may have an arbitrary number of connected components, each with arbitrarily large diameter.