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A finitely generated group can be turned into a geometric object. Choose a finite generating set, build a graph whose vertices are group elements and whose edges connect elements that differ by a generator, and the group becomes a metric space: the distance between two group elements is the length of the shortest word connecting them. This simple encoding—the Cayley graph—is the engine that drives geometric group theory. The subfield's history is the story of increasingly powerful ways to exploit that encoding, moving from combinatorial problems about words to large-scale geometric invariants that classify groups up to quasi-isometry.
The idea of representing a group as a graph dates to Arthur Cayley in the late nineteenth century, but it was Max Dehn in the early 1900s who turned the Cayley graph into a tool for solving algebraic problems. Dehn formulated three decision problems for finitely presented groups: the word problem (decide whether a word equals the identity), the conjugacy problem, and the isomorphism problem. He recognized that the geometry of the Cayley graph—its loops, its dead ends, its shape at large scales—could be used to design algorithms. The word metric, which measures distance along the Cayley graph, made precise the intuition that group elements far from the identity are those requiring long words to express. This framework did not merely illustrate groups; it transformed combinatorial group theory into a subject where geometric intuition could guide algebraic discovery. Cayley graphs and word metrics remain the foundational infrastructure for every later framework in the subfield.
By the 1960s, group theorists had developed powerful algebraic tools for decomposing groups: free products with amalgamation and HNN extensions. These constructions allowed a complicated group to be built from simpler pieces, but the geometric picture was unclear. Bass–Serre theory, developed by Hyman Bass and Jean-Pierre Serre around 1970, supplied that picture. The key insight was that a group acting on a tree—a simply connected graph without cycles—could be decomposed into a graph of groups, where vertex groups correspond to stabilizers of vertices and edge groups to stabilizers of edges. The tree itself is a CAT(0) space (a space of non-positive curvature), so Bass–Serre theory can be seen as the first instance of a group acting geometrically on a non-positively curved space. This framework extended the geometric program beyond the flat Cayley graph: instead of studying a group through its action on its own Cayley graph, one studies the group through its action on a different, often simpler, geometric object. Bass–Serre theory coexists with later curvature frameworks as a specialized tool for detecting splittings, but its tree-based decompositions are now understood as a special case of the broader CAT(0) framework.
The 1980s brought a decisive shift in perspective. Instead of asking what a group looks like up close, coarse geometry asks what it looks like from far away. The central equivalence relation is quasi-isometry: two metric spaces are quasi-isometric if their large-scale shapes are the same, ignoring bounded distortions. For finitely generated groups, the quasi-isometry type of the Cayley graph is independent of the choice of finite generating set, so quasi-isometry becomes a group invariant. This framework, developed by Mikhael Gromov and others, introduced a suite of coarse invariants: growth (polynomial, exponential, intermediate), number of ends, asymptotic dimension, amenability, and the property of being hyperbolic. Coarse geometry did not replace Cayley graphs; it provided the language and the conceptual infrastructure that made the curvature-based frameworks of the same decade possible. Without the coarse perspective, the idea that a group could be classified by its large-scale curvature would have made little sense.
If a group acts properly and cocompactly by isometries on a complete geodesic space that satisfies the CAT(0) inequality—a condition that generalizes non-positive curvature from Riemannian manifolds to metric spaces—then the group inherits strong geometric and algebraic properties. CAT(0) groups, studied systematically by Gromov and later by many others, include free abelian groups, surface groups, and many groups arising in low-dimensional topology. The CAT(0) condition is a direct generalization of the tree geometry that underlies Bass–Serre theory: every tree is a CAT(0) space, so Bass–Serre decompositions are a special case. But CAT(0) spaces are far richer: they can contain flats (isometric copies of Euclidean space), which introduce complications absent in trees. The presence of flats means that CAT(0) groups need not be hyperbolic; they occupy a broader, more flexible landscape. CAT(0) groups remain an active research area, with open questions about which groups admit such actions and how the CAT(0) boundary behaves.
In 1987, Gromov published a landmark essay that defined a group to be hyperbolic if its Cayley graph is δ-hyperbolic: every geodesic triangle is δ-thin, meaning its sides are uniformly close to each other. This condition captures strict negative curvature at large scales. Hyperbolic groups form a subclass of CAT(0) groups when the CAT(0) space has no flats, but the hyperbolic framework is far more than a special case. Gromov showed that hyperbolic groups have solvable word problem (via Dehn's algorithm), a well-defined boundary at infinity (the Gromov boundary), and strong properties about their subgroups and growth. The framework unified many earlier examples—free groups, surface groups, fundamental groups of closed hyperbolic manifolds—and connected group theory to low-dimensional topology, dynamical systems, and analysis on metric spaces. Hyperbolic groups and CAT(0) groups agree on relying on coarse curvature and geometric actions, but they disagree on the strictness of that curvature: hyperbolic groups forbid flats, while CAT(0) groups allow them. This difference has consequences for boundary behavior, algorithmic properties, and the kinds of subgroups that can appear.
Today, the two curvature-based frameworks—hyperbolic groups and CAT(0) groups—are the leading approaches for studying groups through geometry. They share a common foundation in coarse geometry: both use quasi-isometry invariants, both rely on actions on geodesic spaces, and both treat curvature as a large-scale property rather than a local one. Where they diverge is in the strictness of the curvature condition. Hyperbolic groups offer stronger algorithmic guarantees (a solvable word problem via Dehn's algorithm, a computable boundary) and a tighter connection to low-dimensional topology, but they exclude many groups that arise naturally in geometry, such as ℤⁿ for n ≥ 2. CAT(0) groups include these groups and many others, but the CAT(0) condition is harder to verify and the boundary is less well-behaved. Bass–Serre theory persists as a decomposition tool within both frameworks: a hyperbolic group that splits over a subgroup often decomposes along a tree, and the tree is itself a CAT(0) space. Cayley graphs and word metrics remain the universal infrastructure, the starting point for every geometric investigation of a group. The frameworks do not compete for supremacy; they partition the space of groups by curvature, with hyperbolic groups occupying the strictly negative curvature region and CAT(0) groups occupying the broader non-positive curvature region, while coarse geometry provides the common language that makes the comparison possible.