Surface subgroups

A surface group is the fundamental group of a closed surface of non-positive Euler characteristic.    A great deal of recent energy in geometric group theory has focussed on finding surface subgroups in various classes of groups of geometrical interest, especially word-hyperbolic groups. I will survey recent developments, highlights of which include Kahn—Markovic’s solution of the Surface Subgroup conjecture for Kleinian groups and Calegari—Walker’s discovery of surface subgroups in random groups

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