Geometric group theory

Geometric group theory studies groups by examining their actions on geometric spaces, especially how algebraic properties appear as geometric features.

Group action: A group action is a rule assigning each group element a transformation of a set while preserving the group operation. It is the basic way a group acts on a space.
Milnor–Švarc lemma: The Milnor–Švarc lemma states that a group acting properly and cocompactly by isometries on a geodesic space is quasi-isometric to that space. It links a group's word metric to the geometry of a space it acts on.
Max Dehn: Max Dehn was a German mathematician whose work introduced foundational problems about group presentations and surfaces. His decision and conjugacy problems became central geometric group theory questions.
Word problem for groups: The word problem asks whether a word in group generators represents the identity element. Geometry can yield algorithms deciding whether group words are trivial.
Representation theory: Representation theory studies algebraic structures through their linear actions on vector spaces. It studies group actions too, but emphasizes linear algebra rather than coarse space geometry.
Cayley graph: A Cayley graph encodes a group using vertices for elements and edges for multiplication by chosen generators. It turns generators and relations into a geometric space.
Hyperbolic group: A hyperbolic group is a finitely generated group whose Cayley graph is hyperbolic in the sense of thin geodesic triangles. Thin triangles give a geometric condition with strong consequences for group structure.
Dehn's lemma: Dehn's lemma states that a simple closed curve on the boundary of a three-dimensional manifold that bounds a disk also bounds an embedded disk. Dehn's geometric methods helped shape the study of groups through topology.
Dehn function: A Dehn function measures the area needed to fill null-homotopic loops in a presentation's associated complex. It quantifies the geometric complexity of proving relations from a presentation.
Lie group: A Lie group is a group that is also a smooth manifold, with compatible multiplication and inversion. Lie groups carry continuous geometry intrinsically, unlike discrete groups studied through external spaces.

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