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Louis Rowen, Mina Teicher, Uzi Vishne
July 27, 2005

### Abstract

We study Coxeter groups from which there is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set T of transpositions. These quotients, denoted here by C Y ( T ), are a special case of the generalized Coxeter groups defined in [5], and also arise in the computation of certain invariants of surfaces. We use a surprising action of S n on the kernel of the surjection C Y ( T ) → S n to show that this kernel embeds in the direct product of n copies of the free group π 1 ( T ), except when T is the full set of transpositions in S 4 . As a result, we show that each group C Y ( T ) either is virtually Abelian or contains a non-Abelian free subgroup.

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Andrea Lucchini
July 27, 2005

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Angelina Y. M. Chin
July 27, 2005

### Abstract

Let G be a group. A set X of elements of G is said to be non-commuting if xy ≠ yx for any x, y ∈ X with x ≠ y . If | X | ≥ | X′ | for any other non-commuting set X′ in G , then X is said to be a maximal non-commuting set. In this paper we obtain upper and lower bounds for the cardinality of a maximal non-commuting set in an extraspecial p -group where p is an odd prime.

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Lawrence E. Wilson
July 27, 2005

### Abstract

We provide a new proof of the recent result that the torsion elements form a subgroup in certain p -adic analytic groups. In particular, if p is an odd prime and G is a finitely generated p -adic analytic group such that γ h(p–1) (G) ≤ G p [h] , then the torsion elements form a subgroup. This result is best possible as there is a finitely generated p -adic analytic group in which γ h(p-1)+1 (G) ≤ G p [h] for all h ≥ 1 and in which the torsion elements do not form a subgroup. Our proof uses the techniques of pro- p groups and involves much less technical detail then the original proof (though we must borrow one result from that proof ). As part of the proof we also find more information on the elements of finite order in the automorphism group of a uniformly powerful pro- p group.

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Gunnar Traustason
July 27, 2005

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G. Conner, K. Spencer
July 27, 2005

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Let H be the classical Hawaiian earring, and let G be its fundamental group, the Hawaiian earring group. We show that the structure of homomorphisms from the Hawaiian earring group to finite groups behaves in anomalous ways. In particular, there are uncountably many surjective homomorphisms of the Hawaiian earring group to any finite group which cannot be represented by continuous maps.

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Julia M. Wilson
July 27, 2005

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Croke and Kleiner [5] gave a construction for a family { X α : 0 < α ≤ π /2} of CAT(0) spaces that each admit a geometric action by the same group G . They showed that ∂ X α ≉ ∂ X π/2 for all α < π /2. We show that in fact ∂ X α ≉ ∂ X β for all α ≠ β , so that G is a CAT(0) group with uncountably many non-homeomorphic boundaries.

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Murray Elder, Susan Hermiller
July 27, 2005

### Abstract

In this article we show that the Baumslag–Solitar group BS(1, 2) is minimally almost convex, or MAC. We also show that BS(1, 2) does not satisfy Poénaru’s almost convexity condition P (2), and hence the condition P (2) is strictly stronger than MAC. Finally, we show that the groups BS(1, q ) for q ≥ 7 and Stallings’ non-FP 3 group do not satisfy MAC. As a consequence, the condition MAC is not a commensurability invariant.