"Geometric Group Theory" by Matt Clay, U. of Arkansas
"This article is intended to give an idea about how
the topology and geometry of a space influences
the algebraic structure of groups that act on it and
how this can be used to investigate groups."
— Notices homepage summary
A more precise description of the subject . . .
"The key idea in geometric group theory is to study
infinite groups by endowing them with a metric and
treating them as geometric spaces."
— AMS description of the 2018 treatise
Geometric Group Theory , by Drutu and Kapovich
See also "Geometric Group Theory" in this journal.
The sort of thing that most interests me, finite groups
acting on finite structures, is not included in the above
description of Clay's article. That description only
applies to topological spaces. Topology is of little use
for finite structures … unless they are embedded* in
larger spaces that are continuous, not discrete.
* As, for instance, the fifty-six 3-subsets of an 8-set are
embedded in the continuous space of The Eightfold Way .