Metric properties of Houghton's groups

Abstract

Houghton groups were defined in the 1960s, and provided examples of groups which belong to class \(FP_n\) but not to class \(FP_{n+1}\). Recently these groups have been studied from the modern geometric point of view. I will present here some results related to Houghton's groups: find an estimate for their metric, find their automorphism group and their commensurator group, proving that it is infinitely generated. As it embeds into the quasi-isometry group, this provides a large quantity of examples of quasi-isometries of Houghton's groups.