Geometric Group Theory
There are some interesting materials in geometric group theory when I am doing research in computation of Ollivier-Ricci curvature of Cayley graphs:
Terrace Tao’s blog: he builds some analogies of geometric structure between group and manifold, including abelianness–flatness (zero curvature), subgroups–foliations, normal subgroups–bundles…
John Meier’s little blue LMS book: Groups, Graphs, and Trees: An Introduction to the Geometry of Infinite Groups. The book “proceeds step by carefully placed step, quite rapidly and naturally through many of the key ideas of modern geometric group theory,” reviewed by Jon McCammond, UCSB. The study of geometry of finitely generated infinite group is one focus of the theory, and Gromov’s introducing word metric is of course a very insightful way of approaching the geometry. I first read about that in his Riemannian and Non-Riemannian book and also found it in a UChi REU paper “Geometric Group Theory” by Alex Manchester.