"Geometry and Analysis on Groups" Research Seminar
The status of the quasiisometry classification of RAAGs, part II.
Christopher Cashen (Wien)
A right-angled Artin group (RAAG) is defined by a finite simple graph by taking one generator for each vertex and declaring that two generators commute if the corresponding vertices are connected by an edge. Such groups have been studied for a long time, but have recently gained prominence due to work of Wise and Agol connecting them to famous questions from 3-dimensional topology.
An old result of Droms says two RAAGs are isomorphic if and only if their defining graphs are isomorphic. Free groups of different rank already show that this result fails if we replace 'isomorphic' by 'quasiisometric'. However, recent work of Huang shows that there are very strong forms of quasiisometric rigidity present when the defining graphs are sufficiently complicated. Very recent work of Margolis shows how to adapt techniques of Cashen and Martin to answer the quasiisometry problem in cases where the RAAGs split over cyclic subgroups into pieces fitting Huang's conditions. We will survey these results.
The introductory talk will be an introduction to RAAGs and their automorphisms.