We say that a group $G$ is commensurator rigid if it is equal to its own abstract commensurator. There is a pleasing blueprint for proving commensurator rigidity of a group, which runs like so:
1) Find a graph whose isometry group is exactly $G$,
2) Show that the action of $G$ on this graph extends to an action of Comm($G$),
3) Profit!
This method was used recently in proofs by Horbez and myself, and Guerch, to prove commensurator rigidity of outer automorphism groups of free groups and universal Coxeter groups, respectively (subject to necessary rank restrictions in each case). In this talk I will expand on the brief blueprint given above, and discuss forthcoming work with Bridson, where we show that Aut($F_N$) is commensurator rigid when $N$ is at least $3$. The two main inputs are a classification theorem for direct products of $2N −3$ free groups in Aut($F_N$), and rigidity of a certain subcomplex of the free factor complex.
De Yair Minsky
De Ursula Hamenstädt
De Dawid Kielak
De Radhika Gupta
De Anne Lonjou
De Piotr Przytycki
De Vincent Guirardel
De Federica Fanoni
De Corey Bregman
De Richard D. Wade
De Sebastian Hensel
De Stephen Coombes , Peter Liddle
De Ariane Trescases
De Albert Schwarz
De Stephen Coombes , Peter Liddle
De Victor Kleptsyn
De François-Xavier Vialard
