We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the Johnson kernel for such surfaces are naturally isomorphic to the extended mapping class group.
"Automorphisms of the Torelli complex and the complex of separating curves." J. Math. Soc. Japan 63 (2) 363 - 417, April, 2011. https://doi.org/10.2969/jmsj/06320363