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Geometric Group Theory (Definition)

The study of finitely generated groups via the geometry of the spaces on which they act.
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The statement of the theorem

Let Γ\Gamma be a finitely generated group, and let SS be a finite generating set for Γ\Gamma. Consider the Cayley graph Cay(Γ,S)\text{Cay}(\Gamma, S) equipped with the word metric dCayd_{\text{Cay}}. Geometric Group Theory is the study of groups Γ\Gamma that admit a proper, cocompact action on a metric space XX satisfying certain geometric constraints. Specifically, we focus on the following setup:\n\n1. **Action Setup:** A continuous action ρ:Γ↷X\rho: \Gamma \curvearrowright X on a metric space (X,dX)(X, d_X).\n2. **Properness:** The action is proper if for every pair of compact subsets K1,K2⊂XK_1, K_2 \subset X, the set ρ−1(K1∩K2)\rho^{-1}(K_1 \cap K_2) is compact. (This ensures the action is locally controlled).\n3. **Cocompactness:** The action is cocompact if the quotient space X/ΓX/\Gamma is compact. (This implies that the group Γ\Gamma controls the geometry of XX up to a compact set).\n\nWhen XX is a CAT(0)\text{CAT}(0) space (a simply connected space with non-positive sectional curvature), the theory investigates the relationship between the algebraic structure of Γ\Gamma and the geometric properties of XX. A key result is that Γ\Gamma is quasi-isometric to XX. Formally, Γ\Gamma is studied via its quasi-isometry class [Γ]=[(Cay(Γ,S),dCay)]QI[\Gamma] = [(\text{Cay}(\Gamma, S), d_{\text{Cay}})]_{\text{QI}}. The core objective is to classify groups Γ\Gamma based on the geometric properties of their quasi-isometry type, such as being hyperbolic (i.e., Cay(Γ,S)\text{Cay}(\Gamma, S) is a Gromov hyperbolic space) or admitting an action on a CAT(0)\text{CAT}(0) space.