Geometric Group Theory

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5 domain-specific abstractions whose origin domain is Geometric Group Theory.

• Paradoxical set — A set that can be partitioned into finitely many pieces and moved by a group action into two disjoint reconstructions of the whole, exposing nonamenability and the failure of finitely additive invariant size on all subsets.
• Serre's Property FA — A group property requiring every action on a simplicial tree to have a global fixed vertex, equivalently excluding the quotient, amalgam, and ascending-union obstructions identified by Serre.
• Small cancellation theory — The study of group presentations whose relators have sufficiently short mutual overlaps, yielding strong geometric and algorithmic consequences.
• Triangle group — A reflection group generated by the sides of a spherical, Euclidean or hyperbolic triangle with angles π/l, π/m and π/n, acting on the corresponding tessellation.
• Word metric — A left-invariant distance on a generated group equal to the shortest length of a generator word representing one element's difference from another.