Pure Maths Colloquium: Alejandra Garrido

The group of automorphisms of an infinite locally finite graph can be given a totally disconnected locally compact topology with respect to which multiplication and inversion are continuous operations. In other words, it is a totally disconnected locally compact group. If this graph is, for instance, a regular tree, then its group of automorphisms is moreover (almost) simple and generated by a compact set.

In order to understand general locally compact groups, there has been a push in recent years to try to understand, and build more examples of, locally compact groups that are totally disconnected, compactly generated, simple and not discrete. As well as the automorphism group of an infinite regular tree, another typical example of this sort is the group of almost automorphisms of that tree (a.k.a. Neretin’s group). This last group turns out to also be an example of a piecewise full group (a.k.a topological full group) of homeomorphisms of the Cantor set (the boundary of the tree). These piecewise full groups have been a source of new examples of finitely generated infinite simple groups. They are usually built out of certain groupoids, but in this context it is much easier to see them as coming from certain inverse semigroups of partial homeomorphisms of the Cantor set.

After introducing the necessary notions, I will report on joint ongoing work with Colin Reid and David Robertson in which we show how to build compactly generated, simple, totally disconnected locally compact groups out of certain groups, or inverse semigroups, acting on the Cantor set.