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
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.
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