Explicit Constructions of Quasi-Uniform Codes from Groups

We address the question of constructing explicitly quasi-uniform codes from groups. We determine the size of the codebook, the alphabet and the minimum distance as a function of the corresponding group, both for abelian and some nonabelian groups. Potentials applications comprise the design of almost affine codes and non-linear network codes

Groups with frames of translates

Let $G$ be a locally compact group with left regular representation $\lambda_{G}.$ We say that $G$ admits a frame of translates if there exist a countable set $\Gamma\subset G$ and $\varphi\in L^{2}(G)$ such that $(\lambda_{G}(x) \varphi)_{x \in\Gamma}$ is a frame for $L^{2}(G).$ The present work aims to characterize locally compact groups having frames of translates, and to this end, we derive necessary and/or sufficient conditions for the existence of such frames. Additionally, we exhibit surprisingly large classes of Lie groups admitting frames of translates

A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups

The aim of this brief note is to offer another proof of a theorem of Hrushovski that approximate subgroups of GL_n(C) are almost nilpotent. This approach generalizes to uniformly non amenable groups.Comment: 5 page

Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture

Let $G$ be a locally finite group and $F(G)$ the Hirsch--Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G/F(G)$ in $G$. Assume that there is a number $k$ such that the length of every chain of nested centralizers in $G$ does not exceed $k$. The Borovik--Khukhro conjecture states, in particular, that under this assumption the quotient $G/S$ contains an abelian subgroup of index bounded in terms of $k$. We disprove this statement and prove some its weaker analog

Homomorphisms between diffeomorphism groups

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1- manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if M is any closed manifold, and Diff(M)_0 injects into the diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the answer is yes, even under more general conditions. Several lemmas on subgroups of diffeomorphism groups are of independent interest, including results on commuting subgroups and flows.Comment: Contains corrections and additional references. A revised version will appear in Ergodic Theory and Dynamical System

Impartial avoidance and achievement games for generating symmetric and alternating groups

We study two impartial games introduced by Anderson and Harary. Both games are played by two players who alternately select previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.Comment: 12 pages. 2 tables/figures. This work was conducted during the third author's visit to DIMACS partially enabled through support from the National Science Foundation under grant number #CCF-1445755. Revised in response to comments from refere