TITLE: Geometric group theory and analysis
QUESTION [9 upvotes]: Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geometric/topological. Is there any subfield of GGT where methods from analysis find applications? I once heard that analytical tools, e. g. geodesic flows, are used in studying ends of groups, but that's all I know.

REPLY [5 votes]: Not sure why this is not being mentioned but one prominent use of analysis in geometric group theory is in studying boundaries of groups (towards Cannon's conjecture and beyond). See for example discussion here and here.