Hopficity and Co-Hopficity in Soluble Groups

Quasi-isometric Co-hopficity of Non-uniform Lattices in Rank-one Semi-simple Lie Groups

We prove that if G is a non-uniform lattice in a rank-one semisimple Lie group = Isom(H2 R ), then G is quasi-isometrically co-Hopf. This means that every quasi-isometric embedding G → G is coarsely surjective and thus is a quasi-isometry.

Hopfian and co-hopfian subsemigroups and extensions

This paper investigates the preservation of hopficity and co-hopficity on passing to finite-index subsemigroups and extensions. It was already known that hopficity is not preserved on passing to finite Rees index subsemigroups, even in the finitely generated case. We give a stronger example to show that it is not preserved even in the finitely presented case. It was also known that hopficity is...

Expansion in Soluble Groups

Soluble omega-categorical groups

Isoperimetric inequalities for soluble groups

We approach the question of which soluble groups are automatic. We describe a class of nilpotent-by-abelian groups which need to be studied in order to answer this question. We show that the nilpotent-by-cyclic groups in this class have exponential isoperimetric inequality and so cannot be automatic.