On Embeddings of Countable Generalized Soluble Groups into Two-Generated Groups

Groups with no coarse embeddings into hyperbolic groups

We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is “admitting exponentially many fat bigons”, and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct...

Frames Generated by Actions of Countable Discrete Groups

We consider dual frames generated by actions of countable discrete groups on a Hilbert space. Module frames in a class of modules over a group algebra are shown to coincide with a class of ordinary frames in a representation of the group. This has applications to shift-invariant spaces and wavelet theory. One of the main findings in this paper is that whenever a shift-invariant subspace in L2(R...

A Class of Two Generated Automaton Groups on a Three Letter Alphabet

Two-generated Groups Acting on Trees

We study 2-generated subgroups of groups that act on simplicial trees. We show that any generating pair {g, h} of such a subgroup is Nielsen-equivalent to a pair {f, s} where either powers of f and s or powers of f and sfs−1 have a common fixed point if the subgroup 〈g, h〉 is freely indecomposable. Analogous results are obtained for generating pairs of fundamental groups of graphs of groups. So...

On Infinite Groups Generated by Two Quaternions

Let x, y be two integer quaternions of norm p and l, respectively, where p, l are distinct odd prime numbers. What can be said about the structure of 〈x, y〉, the multiplicative group generated by x and y ? Under a certain condition which excludes 〈x, y〉 from being free or abelian, we show for example that 〈x, y〉, its center, commutator subgroup and abelianization are finitely presented infinite...