a hyperbolic group
Some groups behave like the symmetries of a negatively curved world — like the fundamental group of a surface with many handles, or a free group, where everything spreads apart and there are no big flat regions. A hyperbolic group is a finitely generated group whose geometry, seen through its Cayley graph, is negatively curved in the large-scale sense. These groups are simultaneously the best-understood class of infinite groups and the 'generic' one, and they have algorithmically tame structure.
The definition transports Gromov-hyperbolicity from spaces to groups. A finitely generated group G is hyperbolic if its Cayley graph (with respect to some, equivalently any, finite generating set) is a Gromov-hyperbolic metric space — its geodesic triangles are uniformly delta-thin. Because hyperbolicity of a space is a quasi-isometry invariant and different generating sets give quasi-isometric Cayley graphs, this does not depend on the chosen generators, so 'hyperbolic' is a property of the group itself. Equivalently, G is hyperbolic exactly when it acts geometrically on some Gromov-hyperbolic space (via the Švarc-Milnor lemma). Free groups, surface groups of genus >= 2, and fundamental groups of compact negatively curved manifolds are all hyperbolic; Z^2 is not, because its grid Cayley graph has arbitrarily fat triangles.
Hyperbolic groups are remarkably well-behaved. They have a solvable word problem in linear time and a solvable conjugacy problem — Dehn's algorithm works because thin triangles force any word representing the identity to contain a large chunk of a defining relator that can be shortened. They have finitely many conjugacy classes of torsion, are finitely presented, and carry a boundary at infinity whose topology constrains them. Two honest caveats. First, hyperbolic is a large-scale, coarse property: it says nothing about the finite group structure and is invariant under passing to finite-index subgroups or quotienting finite normal subgroups. Second, Z^2 and more generally any group containing a copy of Z^2 (a 'flat') is NOT hyperbolic — the presence of an embedded large flat is exactly what hyperbolicity forbids, so being hyperbolic is incompatible with having Z + Z inside.
The fundamental group of a closed orientable surface of genus 2 is hyperbolic. That surface carries a hyperbolic metric (constant curvature -1), so its universal cover is the hyperbolic plane H^2, on which the group acts geometrically by deck transformations; by Švarc-Milnor the group is quasi-isometric to H^2, which is Gromov-hyperbolic, so the group is hyperbolic. Concretely its Cayley graph has uniformly thin triangles. The torus group Z^2, by contrast, acts on flat R^2 and is not hyperbolic.
Genus-2 surface group acts on H^2 and is hyperbolic; the torus group Z^2 acts on flat R^2 and is not.
A group containing Z^2 cannot be hyperbolic — an embedded flat plane is exactly what thin triangles forbid. So 'negatively curved' for groups means no large flats, not merely 'not Euclidean'; Z^2 itself is the basic non-example.