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Geometric group theory has natural connections and rich interfaces with many of the other major fields of modern mathematics. The basic motif of the field is the construction and exploration of actions by infinite groups on spaces that admit further structure, with an emphasis on geometric structures of different sorts: one usually seeks actions in order to illuminate the structure of groups of particular interest, but one also explores actions in order to understand the underlying spaces. The dramatic growth of the field in the late twentieth century was closely associated with the study of generalized forms of non-positive and negative curvature, and classically the spaces at hand were cell complexes with some additional structure. But the scope of the field, the range of groups embraced by its techniques, and the nature of the spaces studied, have expanded enormously in recent years, and they continue to do so. This meeting provided an exciting snapshot of some of the main strands in the recent development of the subject.
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Martin R. Bridson, Linus Kramer, Bertrand Rémy, Karen Vogtmann, Geometric Structures in Group Theory. Oberwolfach Rep. 14 (2018), no. 2, pp. 1869–1915