群的端与双曲群

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.

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.