hyperbolic group
In the flat plane a triangle can be fat; its sides bow apart and the middle is far from any edge. In a negatively curved world — think of a saddle, or the hyperbolic plane — triangles are always skinny: each side hugs close to the other two, so the whole triangle is ‘thin.’ A hyperbolic group is one whose Cayley graph behaves like that negatively curved world: every geodesic triangle is thin, uniformly, no matter how big.
Made precise by Gromov, a geodesic metric space is δ-hyperbolic if there is a constant δ ≥ 0 such that in every geodesic triangle each side lies in the δ-neighborhood of the union of the other two (the thin-triangles condition). A finitely generated group is hyperbolic if some, equivalently any, Cayley graph is δ-hyperbolic for some δ; this is a quasi-isometry invariant, so it is a property of the group, not the chosen generators.
Hyperbolic groups are abundant and beautifully behaved. They are finitely presented, have solvable word and conjugacy problems (with a linear-time, automatic word problem), satisfy a linear isoperimetric — that is, linear Dehn function — inequality, and obey the Tits alternative: every subgroup is virtually cyclic or contains a free group of rank 2. Examples include all finite groups, free groups, surface groups of genus at least 2, and fundamental groups of closed negatively curved manifolds; in a precise probabilistic sense, a ‘random’ finitely presented group is hyperbolic.
The fundamental group of a closed orientable surface of genus 2, ⟨ a, b, c, d | [a,b][c,d] ⟩, is hyperbolic: it acts geometrically on the hyperbolic plane, whose triangles are uniformly thin. By contrast Z^2 is not hyperbolic — its flat grid has arbitrarily fat triangles.
Genus-2 surface groups are hyperbolic; the flat group Z^2 is not.
Whether every hyperbolic group is residually finite is a famous open problem; a yes would follow from (and is essentially equivalent to) every hyperbolic group being virtually special in Agol-Wise theory for the cubulated case.