Geometric Group Theory: An Introduction (Universitext)
Features more than 250 exercises of varying difficultyincluding programming tasksIntroduces the key notions from quasi-geometry, such as growth,hyperbolicity, boundary constructions and amenabilityAssumes only a basic background in group theory, metric spacesand point-set topology——————————Inspired by classical geometry, geometric group theory has inturn provided a variety of applications to geometry, topology,group theory, number theory and graph theory. This carefullywritten textbook provides a rigorous introduction to this rapidlyevolving field whose methods have proven to be powerful tools inneighbouring fields such as geometric topology.Geometric group theory is the study of finitely generated groupsvia the geometry of their associated Cayley graphs. It turns outthat the essence of the geometry of such groups is captured in thekey notion of quasi-isometry, a large-scale version of isometrywhose invariants include growth types, curvature conditions,boundary constructions, and amenability.This book covers the foundations of quasi-geometry of groups atan advanced undergraduate level. The subject is illustrated by manyelementary examples, outlooks on applications, as well as anextensive collection of exercises.