2 LEE MOSHER, MICHAH SAGEEV, AND KEVIN WHYTE

1.1. Example applications. First, here is a theorem regarding two funda-

mental groups of hyperbolic manifolds amalgamated along infinite cyclic subgroups.

For the proof see Theorem 7.1, which relies essentially on a theorem of Schwartz

[Sch97].

Theorem 1.1. Let A and B be fundamental groups of closed hyperbolic man-

ifolds of dimension at least 3, and let G be the amalgamated free product A ∗Z B

where Z includes as a maximal cyclic subgroup in both A and B. If H is a finitely

generated group quasi-isometric to G then H splits as a graph of groups each of

whose vertex groups is commensurable to A or B. In particular, there are infinitely

many quasi-isometry classes of groups of this form.

In fact, in the context of Theorem 1.1 one obtains a complete computation of

the quasi-isometry group; see Proposition 7.13.

Notice that in the above example, the vertex-to-edge inclusions are codimension

2. As we will see below, Alexander duality can be applied in a straightforward

manner. In the codimension-1 setting, one typically needs that the edge subgroups

in a vertex group cross one another in some suitable sense. Here is one such example.

A collection of geodesics in the hyperbolic plane is said to be filling if the

complementary regions are all bounded. A collection of cyclic subgroups in a surface

group is said to be filling if the pattern of geodesics in the universal cover is filling.

Theorem 1.2. Suppose that G splits as a graph of groups where each vertex

group is a surface group, each edge group is a cyclic group and the edge-to-vertex

inclusions provide a filling collection of curves on the vertex group. Then any

torsion-free group quasi-isometric to H splits as a graph of groups whose edge groups

are cyclic and whose vertex groups are surface groups or cyclic groups.

A stronger version of Theorem 1.2 is given in Theorem 7.8, with commensura-

bility information built into the conclusion, by applying a theorem of Kapovich and

Kleiner [KK00]. Using this we will obtain infinitely many distinct quasi-isometry

classes of groups of this type.

The next theorem gives examples with higher dimensional edge groups, namely

fundamental groups of surfaces. For the proof see Theorem 7.4, and for even

stronger rigidity conclusions in many examples see Theorem 7.14.

Theorem 1.3. Let A be the fundamental group of a hyperbolic 3-manifold which

fibers over the circle in two ways, with fibers F1 and F2. Let φ be any isomorphism

between (finite index subgroups of) π1(F1) and π1(F2) and define G as the HNN

extension A∗φ. If H is any torsion free group quasi-isometric to G, then H splits as

a graph of groups, whose edge groups are surface groups, and whose vertex groups

are hyperbolic 3-manifold groups commensurable to A and surface groups.

The commensurability information in Theorem 1.3 is an application of a theo-

rem of Farb and Mosher [FM02b].

Here is a theorem for abelian groups.

Theorem 1.4. Suppose A is a finitely generated abelian group of rank n,

G1,G2 A are two rank n − 1 subgroups that span A, and G is an HNN-extension

identifying G1 to G2. Then any group H quasi-isometric to G splits as a graph of

virtually abelian groups with edge groups of rank n − 1 and vertex groups of rank at

most n.