David Hume

Assistant Professor, University of Birmingham

My current research is in geometric group theory: a young, diverse, active and fascinating branch of mathematics at the intersection of algebra, geometry, combinatorics and topology. The main goal of geometric group theory is to reveal new relationships between algebraic properties of groups and the geometric properties of spaces they act on, and to use these techniques to answer important problems in adjacent areas of mathematics and computer science. A particularly rich vein of work in geometric group theory focusses on the algebraic properties two groups must share if they act nicely on metric spaces which are sufficiently similar "at large scales". Such properties include admitting a finite presentation and possessing a finite-index nilpotent subgroup.

This approach becomes problematic when we instead wish to study the subgroups of a group, since the metric spaces that the subgroup and group act on need not be similar at large scales, instead there will be an inclusion of the space corresponding to the subgroup into the space corresponding to the group which is "injective at large scales". Many of the properties which are known to be preserved for groups acting on similar metric spaces do not pass from groups to their subgroups - and indeed, until recently there were very few such tools. The goal of my research is to introduce and determine new tools which are inherited by subgroups.

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