Differential geometry on finitely presented groups and related combinatorial invariants

Many concepts in differential geometry (for example, Riemannian metric, length, shortest arc, volume), defined originally for manifolds, are easily transferred to simplicial polyhedra. This extends significantly the scope of application of such concepts familiar in the framework of manifolds as systolic volume and volume entropy. Many properties of these invariants depend only on the fundamental group of the polyhedron. Using the process of minimisation, it becomes possible to transfer these invariants directly to finitely presented groups. In studying the systolic area and volume entropy of groups, new combinatorial invariants arise, which are of independent interest. The paper is an introduction to this area at the intersection of geometry, topology, and group theory. Bibliography: 65 titles.