Spaces of graphs and tori

Abstract: The classical symmetric space SL(n,R)/SO(n) is often described as a space of marked lattices in R^n (marked by a choice of basis), but from another perspective it can be thought of as a space of flat n-dimensional tori, marked by a choice of basis for the fundamental group Z^n. In the 1980’s a space O_n, now called Outer space, was defined analogously as a space of marked metric graphs, marked by a choice of basis for the fundamental group F_n. I will describe these spaces and explain some of the ways in which they can be used to give information about automorphisms of Z^n and F_n respectively. I will then show how to make a hybrid space which can be used to give information about automorphisms of groups that interpolate between F_n and Z^n, called `right-angled Artin groups`.

Mathematics

Audience: researchers in the discipline

Barcelona Mathematical Days 2020

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