Ends: directions to infinity
Take the Cayley graph, delete a large ball, and count how many unbounded connected components survive as the ball grows. That stable count is the number of ends of the group — a quasi-isometry invariant, hence intrinsic. The astonishing theorem (Hopf, Freudenthal, Stallings) is that the count is severely limited: a finitely generated group has exactly 0, 1, 2, or ∞ ends, never 3 or 17.
The four cases and what they MEAN:
0 ends <=> the group is finite.
2 ends <=> the group is virtually Z (a finite-index copy of Z).
geometrically: a line, two ways out.
inf ends <=> the group splits as a nontrivial amalgam or HNN
over a finite subgroup (STALLINGS' THEOREM)
e.g. F_2 has infinitely many ends -- its tree frays
into infinitely many branches.
1 end <=> everything else, the 'generic' case
e.g. Z^2 (delete a ball from the plane: one piece remains).
Stallings turns a coarse COUNT into an algebraic SPLITTING.Thin triangles: Gromov hyperbolicity
Now the modern centerpiece. A geodesic metric space is δ-hyperbolic if every geodesic triangle is δ-thin: each side lies within distance δ of the union of the other two. In the Euclidean plane triangles can be fat (no δ works); in a tree they are infinitely thin (δ = 0); the hyperbolic plane sits at finite δ. A finitely generated group is a hyperbolic group if its Cayley graph is δ-hyperbolic for some δ. By the work of guide 3 this is a quasi-isometry invariant, so it is a property of the group, not the generators.
Who is hyperbolic?
YES finite groups (trivially), virtually free groups, F_n (delta = 0),
surface groups of genus >= 2, virtually Z,
'random' finitely presented groups (Gromov: generic => hyperbolic).
NO Z^2 : it contains a flat plane, fat triangles, so NOT hyperbolic.
=> any group containing a Z^2 subgroup is NOT hyperbolic
(Z^2 is a quasi-isometrically embedded flat -- a 'no-go' certificate).
Baumslag-Solitar BS(1,2) is not hyperbolic either.
Rule of thumb: hyperbolic = 'negatively curved' = no flats, tree-like at large scale.Why hyperbolicity is worth so much: it tames everything we struggled with. Hyperbolic groups have solvable [[alg-word-problem|word problem]] in linear time — the Dehn function is linear, so a greedy ‘Dehn's algorithm’ shortens any loop that is trivial. They are finitely presented, have at most exponential growth, and satisfy the Tits alternative: every subgroup either contains a free group of rank 2 or is virtually cyclic — no exotic in-between behavior.