We use geometry of Davis complex of a Coxeter group to investigate finite index reflection subgroups of Coxeter groups. The main result is the following: if G is an infinite indecomposable Coxeter group and H ⊂ G is a finite index reflection subgroup then the rank of H is not less than the rank of G. This generalizes results of [7]. We also describe some properties of the nerves of the group an...

Working over the field Q(t), Kihara constructed an elliptic curve with torsion group Z/4Z and five independent rational points, showing the rank is at least five. Following his approach, we give a new infinite family of elliptic curves with torsion group Z/4Z and rank at least five. This matches the current record for such curves. In addition, we give specific examples of these curves with high...

This is a survey and announcement of recent results on the structure of free profinite semigroups using techniques and results from symbolic dynamics. The intimate connection between uniformly recurrent infinite words and J -maximal regular J -classes is explored to compute the maximal subgroups of the J -classes associated with Arnoux-Rauzy infinite words, which turn out to be free profinite g...

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, it remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existenc...

In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks.

We give axiomatic foundations for infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. Continuing work of Higgs and Oxley, this completes the solution to a problem of Rado of 1966.

Recently, a number of new classes of infinite-dimensional simple Lie algebras over a Field of characteristic 0 were discovered by several authors (see the references at the end of this paper). Among those algebras, are the generalized Witt algebras. The higher rank Virasoro algebras was introduced by Patera and Zassenhaus [PZ], which are 1-dimensional universal central extensions of some genera...

In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decomposit...

In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino ...

The automorphism group of a 3-generated 2-group G of intermediate growth is determined and it is shown that the outer group of automorphisms of G to be an elementary abelian 3group of infinite rank.