A much more general result is given in Corollary 7.11, applying to a broad
spectrum of graphs of abelian groups.
1.2. The methods of proof: a special case. We illustrate some of the ideas
involved in proving our main theorems by sketching the proof in the special case of
theorem 1.1. Let M and N be closed hyperbolic n-manifolds for some n 3. Let
A and B be their fundamental groups. Choosing maximal cyclic subgroups of A
and B amounts to choosing primitive closed geodesics in M and N. Let Y be the
space built by gluing an annulus to M and N, with the boundary circles attached
to the chosen geodesics. The group G = A ∗Z B is π1(Y ), and so acts properly
discontinuously and cocompactly on X = Y . The space X is built from copies of
Hn glued together with strips attached along the geodesics which cover the chosen
curves in M and N. There is a natural G-equivariant map, π : X T , from X to
the Bass-Serre tree T of the splitting of G in which each copy of
is mapped to
a single vertex, and each strip maps to an edge. The is an example of what we call
a Bass-Serre complex or a Bass-Serre tree of spaces, which serves as the standard
model space for our considerations. (See Section 2.4 for the general construction.)
We call the copies of
in X the vertex spaces Xv, one for each vertex v T .
The strips are called the edge spaces Xe, one for each edge e T .
We start by understanding the self quasi-isometries of G. This is typically a
crucial step in proving rigidity results. The salient point is that a quasi-isometry
from some mystery group H to G provides a natural map from H to QI(G), the
quasi-isometry group of G. Thus one seeks geometric patterns which are invariant
under QI(G). As X is quasi-isometric to G, we can work with quasi-isometries of
X. Since X is uniformly contractible, we can move any quasi-isometry a bounded
distance to a continuous map. Let f : X X be such a continuous quasi-isometry.
Our method of proof consists of two main steps.
Step 1 Vertex Rigidity: There is an R 0 such that for any vertex space
Xv X there is a vertex space Xv X with dH(f(Xv),Xv ) R, where dH(·, ·)
denotes the Hausdorff metric.
Step 2 Tree Rigidity: There is a automorphism τ of T and a quasi-isometry
f : X X at bounded distance from f such that π f = τ π.
Conclusion: If we establish steps 1 and 2, then we may conclude that if H
is quasi-isometric to G then H acts on T with finite quotient; for every vertex
v of T , Hv is quasi-isometric to Gv and similarly, for every edge e, He is quasi-
isometric to Ge. In particular, H has a splitting as a graph of groups with virtually
cyclic edge groups and vertex groups quasi-isometric to
A theorem of Schwartz
[Sch97] then tells us that the vertex and edge groups of the splitting for H are
commensurable to the corresponding ones for G.
Let us now see how steps 1 and 2 are proven in this setting.
Vertex rigidity. Suppose that vertex rigidity fails. Then one finds an edge
space Xe such that f(Xv) travels deeply into both complementary components of
Xe. This will implies that Xe (which we recall is quasi-isometric to a line) coarsely
separates f(Xv). But now note that f(Xv) is quasi-isometric to
so we get
a subset of a line coarsely separating
with n 3. This contradicts a coarse
version of Alexander duality.
Tree rigidity. The vertex rigidity step yields a bijection f# : Verts(T )
Verts(T ), so that f(Xv) has finite Hausdorff distance from Xf#(v). We now wish
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