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Ralf Gramlich, Max Horn, Antonio Pasini, Hendrik Van Maldeghem
July 16, 2008

### Abstract

We study geometries that arise from the natural G 2 () action on the geometry of one-dimensional subspaces, of non-singular two-dimensional subspaces, and of non-singular three-dimensional subspaces of the building geometry of type C 3 (), where is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-standard geometric covering theory from [R. Gramlich and H. Van Maldeghem. Intransitive geometries. Proc. London Math. Soc. (2) 93 (2006), 666–692.] is not applicable. In this paper we introduce the concept of fused amalgams in order to extend the geometric covering theory so that it applies to that geometry. This yields an interesting new amalgamation result for the group G 2 ().

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Peter A. Brooksbank, E. A. O'Brien
July 16, 2008

### Abstract

We describe the structure of the subgroup of the general linear group defined over a finite field that preserves two bilinear or sesquilinear forms of the same classical type, at least one of which is non-degenerate. This description underpins an algorithm to construct the intersection of two classical groups of the same type.

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U. Meierfrankenfeld, G. Stroth
July 16, 2008

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Eric Schmutz
July 16, 2008

### Abstract

Let μ n be the average of the orders of the elements the unitary group U( n , q ). The following conjecture of Fulman is proved: for any fixed q , log μ n = n log q − log n + o q (log n ) as n → ∞.

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Primož Moravec
July 16, 2008

### Abstract

In this note we describe the exponent semigroups of finite p -groups of maximal class and finite p -groups of class at most 5. Consequently, sharp bounds for the exponent of the Schur multiplier of a finite p -group of class at most 4 are obtained. Our results extend some well-known results of Jones (1974).

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J. C. Beidleman, H. Heineken, M. F. Ragland
July 16, 2008

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Stefaan Delcroix
July 16, 2008

### Abstract

In this paper, we prove that if G is a non-finitary locally finite, simple group then the following holds: (a) G is of p -type for some prime p if and only if G is not finitary and there exist a prime q ≠ p and x ∈ G such that | x | is a power of q and 〈 x Q 〉 is abelian for all q -subgroups Q of G containing x . (b) G is of alternating type if and only if for any prime p and any x ∈ G with | x | a power of p , there exists a p -subgroup P of G containing x such that 〈 x P 〉 is not solvable.

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Miklós Abért
July 16, 2008

### Abstract

We show that it is consistent with the axioms of set theory that every infinite profinite group G possesses a closed subset X of Haar measure zero such that less than continuum many left translates of X cover G . This answers a question of Elekes and Tóth and by their work settles the problem for all infinite compact topological groups.

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Olivier Siegenthaler
July 16, 2008

### Abstract

Based on the work of Abercrombie [A. G. Abercrombie. Subgroups and subrings of profinite rings. Math. Proc. Cambridge Philos. Soc. 116 (1994), 209–222.], Barnea and Shalev [Y. Barnea and A. Shalev. Hausdorff dimension, pro- p groups, and Kac–Moody algebras. Trans. Amer. Math. Soc. 349 (1997), 5073–5091.] gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree 𝒯. Abért and Virág [M. Abért and B. Virág. Dimension and randomness in groups acting on rooted trees. J. Amer. Math. Soc. 18 (2005), 157–192.] showed that there exist finitely generated (but not necessarily level-transitive) subgroups of Aut 𝒯 of arbitrary dimension in [0, 1]. In this article we explicitly compute the Hausdorff dimension of the level-transitive spinal groups. We then give examples of 3-generated spinal groups which have transcendental Hausdorff dimension, and construct 2-generated groups whose Hausdorff dimension is 1.

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O. Bogopolski, E. Ventura
July 16, 2008

### Abstract

While Dehn functions D ( n ) of finitely presented groups have been well studied in the literature, mean Dehn functions have received less attention. Gromov introduced the notion of the mean Dehn function D mean ( n ) of a group, suggesting that in many cases it should grow more slowly than the Dehn function itself This paper presents computations pointing in this direction. In the case of any finite presentation of an infinite finitely generated abelian group (for which it is well known that D ( n ) ~ n 2 except in the 1-dimensional case), we show that the three variants D osmean ( n ), D smean ( n ) and D mean ( n ) all are bounded above by Kn (In n ) 2 , where the constant K depends only on the presentation (and the geodesic combing) chosen. This improves an earlier bound given by Kukina and Roman'kov.