Mathematics Colloquia and Seminars

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An action of a group on an R-tree is said to be small if the stabilizer of every edge is virtually cyclic. Small actions of hyperbolic groups play a role in low dimensional topology and geometric group theory, since they relate to the theory of measured laminations for surfaces and to the outer automorphism group of free groups, among others. Geodesic currents are a space of measures on the square of the boundary at infinity of the group, with interesting geometric and dynamical properties. In particular, they contain all conjugacy classes of elements of the group. In 1991, Bonahon conjectured that the stable length of a small action of a hyperbolic group extends continuously to geodesic currents. In this talk, I will present the necessary background for this conjecture and sketch a proof. This is work in progress with Misha Kapovich.