|Date||Dec 21, 2022|
A (unmarked, unoriented) length spectrum of a Riemannian manifold is the set of the lengths of all closed geodesics. For compact hyperbolic surfaces, length spectra, Laplacian spectra and hyperbolic metrics are deeply concerned; for example, Selberg trace formula, Wolpert's theorem and Sunada's theorem. Meanwhile, Gromov studied geometric aspects of finitely generated groups, especially, word-hyperbolic groups. The class of word-hyperbolic groups properly includes fundamental groups of all hyperbolic manifolds. In this talk, we are going to see a property about word-length spectra of word-hyperbolic groups and some generalization to surface mapping class groups and free products of groups. This is joint work with Hyungryul Baik and Hyunshik Shin. Keywords: coarse geometry, word-hyperbolic groups, stable translation length, word-length spectrum, surface mapping class groups, free products of groups.