Quasiconformal groups, Patterson-Sullivan theory, and local analysis of limit sets

We extend the part of Patterson-Sullivan theory to discrete quasiconformal groups that relates the exponent of convergence of the Poincaré series to the Hausdorff dimension of the limit set. In doing so we define new bi-Lipschitz invariants that localize both the exponent of convergence and the Hausdorff dimension. We find these invariants help to expose and explain the discrepancy between the conformal and quasiconformal setting of Patterson-Sullivan theory.

     

An addition theorem for finite Abelian groups

If g₁, g₂, …, g_2n?1 is a sequence of 2n ? 1 elements in an Abelian group G of order n, it is known that there are n distinct indices i₁, i₂, …, i_n such that 0 = g_i1 + g_i2 + ? + g_in. In this paper a suitably general condition on the sequence is given which insures that every element g in G has a representation g = g_i1 + g_i2 + ? + g_in as the sum of n terms of the sequence.     

Projections of Patterson-Sullivan measures and the dichotomy of Mohammadi-Oh

Let Γ be some discrete subgroup of SO°(n + 1, R) with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the N component in some Iwasawa decomposition SO°(n+1, R) = KAN. We also study the dimension of projected Patterson-Sullivan measures along some fixed direction.     

A Hall-Higman-Shult type theorem for arbitrary finite groups

We shall extend a fixed point theorem of Shult to arbitrary finite groups. This will have applications to the study of group automorphisms.     

The Tarski irredundant basis theorem and the finite soluble groups

Denote by and , respectively, the smallest and the largest cardinality of a minimal generating set of a finite group G. The Tarski irredundant basis theorem implies that for every k with there exist a minimal generating set , an index and in G such that is again a minimal generating set of G. In this case we say that is an immediate descendant of ω. There are several examples of minimal generating sets of cardinality smaller than which have no immediate descendant and so it appears an interesting problem to investigate under which conditions an immediate descendant exists. In this paper we discuss this problem in the case of finite soluble groups.     

The Chevalley-Warning theorem and a combinatorial question on finite groups

Recently, W. D. Gao (1996) proved the following theorem: For a cyclic group of prime order, and any element in it, and an arbitrary sequence of elements from , the number of ways of writing as a sum of exactly of the 's is or modulo according as is zero or not. The dual purpose of this note is (i) to give an entirely different type of proof of this theorem; and (ii) to solve a conjecture of J. E. Olson (1976) by answering an analogous question affirmatively for solvable groups.

     

Algebraic groups over finite fields, a quick proof of Lang's theorem

We give an easy proof of Lang's theorem about the surjectivity of the Lang map on a linear algebraic group defined over a finite field, where is a Frobenius endomorphism.

     

An isolic generalization of Cauchy's theorem for finite groups

In his note [5] Hausner states a simple combinatorial principle, namely:

A new trichotomy theorem for groups of finite Morley rank

There is a longstanding conjecture, of Gregory Cherlin and BorisZilber, that all simple groups of finite Morley rank are simplealgebraic groups. Here we will conclude that a simple K*-groupof finite {M}orley rank and odd type either has normal rankof at most 2, or else is an algebraic group over an algebraicallyclosed field of characteristic not 2. To this end, it sufficesto produce a proper 2-generated core in groups with \Pruferrank 2 and normal rank at least 3, which is what is proved here.Our final conclusion constrains the Sylow 2-subgroups availableto a minimal counterexample and, finally, proves the trichotomytheorem in the nontame context.