Maximal Order Three-Rewriteable Subgroups of Symmetric Groups

Recently, Burns and Goldsmith [2] characterized the maximal order Abelian subgroups of the symmetric groups using elementary techniques and the results of Hoffman [5]. This classification could also be directly inferred from the results of Kovacs and Praeger [7]. A natural extension would be to consider the weaker, more general form of commutativity, three-rewriteability. The purpose of this paper is to completely characterize the maximal order three-rewriteable subgroups of the symmetric groups

A classification of certain maximal subgroups of symmetric groups

Problem 12.82 of the Kourovka Notebook asks for all ordered pairs (n,m) such that the symmetric group Sn embeds in Sm as a maximal subgroup. One family of such pairs is obtained when m=n+1. Kalužnin and Klin [L.A. Kalužnin, M.H. Klin, Certain maximal subgroups of symmetric and alternating groups, Math. Sb. 87 (1972) 91–121] and Halberstadt [E. Halberstadt, On certain maximal subgroups of symmetric or alternating groups, Math. Z. 151 (1976) 117–125] provided an additional infinite family. This paper answers the Kourovka question by producing a third infinite family of ordered pairs and showing that no other pairs exist

Weakly commensurable groups, with applications to differential geometry

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic groups. We have included a discussion of very recent results and conjectures on absolutely almost simple algebraic groups having the same maximal tori and finite-dimensional division algebras having the same maximal subfields.Comment: Improved exposition, updated bibliography. arXiv admin note: substantial text overlap with arXiv:1212.121

Finsler bordifications of symmetric and certain locally symmetric spaces

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X/\Gamma$ for arbitrary discrete subgroups $\Gamma< G$. These bordifications result from attaching $\Gamma$-quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion free case, to a question of Ha\"issinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.Comment: 88 page