We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the second one as their scarcity. We discuss examples for both cases.
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Acknowledgements This work was partly funded by the Leverhulme Trust Research Project Grant RPG-2017-159. MM is supported by the grant Sonatina 2018/28/C/ST1/00542 funded by Narodowe Centrum Nauki. MM and JK were partially supported by SFB 1085 “Higher Invariants” funded by Deutsche Forschungsgemeinschaft.