to see that f carries neighboring vertex spaces to neighboring vertex spaces. The
key point here is that in X, adjacent vertex spaces are characterized coarsely by
the fact that they have neighborhoods with unbounded intersection, whereas any
intersection of neighborhoods of non-adjacent vertex spaces is bounded. This fact
relies on the assumption that the cyclic edge groups are maximal in their respective
vertex spaces. Thus f# carries adjacent vertices to adjacent vertices and hence
extends to an isometry τ of T as required.
1.3. The general setting. The proofs of our main theorems for more general
graphs of groups follow a similar outline. Let us discuss some of the issues that
arise in the more general setting, and the various hypotheses needed to deal with
these issues.
For starters, our graphs of groups G will always satisfy the standard hypothesis
of irreducibility, meaning that for every edge of G with distinct endpoints, both
of its edge-to-vertex injections have image of index 2. Every finite graph of
groups can be made irreducible by inductively collapsing edges, without changing
the fundamental group.
Let T denote the Bass-Serre tree of G, and let X denote the Bass-Serre complex
of G. The group π1G acts on X and T , and there is a π1G equivariant projection
map X T . The action on X is properly discontinuous and cocompact, and the
action on T is cocompact with quotient graph of groups G. The inverse image under
the projection map X T of a vertex v T is a vertex space denoted Xv, and the
inverse image of the midpoint of an edge e T is an edge space denoted Xe.
Vertex Rigidity. The proof of the vertex rigidity step in Section 1.2 above
relied on the fact that no geodesic can coarsely separate Hn when n 3. In
general we want to know that edge spaces cannot disconnect vertex spaces. In
most applications the essential focus is on the maximal vertex spaces, those not
strictly coarsely contained in any other vertex space. We call these the depth zero
vertex spaces. An important hypothesis of our results will be that every vertex
space is coarsely contained in a depth zero vertex space.
To see that edge spaces cannot coarsely separate the depth zero vertex spaces we
generally use a coarse version of Alexander duality to understand the separation
properties of subsets. The right context for these arguments is that of coarse
Poincare duality spaces introduced in [KK99]. Almost all of our results assume
that the depth zero vertex spaces are coarse PD spaces (see section 2.6 for detailed
definitions). To put it another way, the stabilizer groups of depth zero vertices are
required to be “coarse PD groups”. This is an essential hypothesis for all of our
There are several different vertex rigidity phenomena. For example, given a
coarse PD(n) vertex space, if the incident edge spaces are of dimension at most
n−2 then we can use a version of the vertex rigidity step of Section 1.2: in a coarse
PD(n) space, no subspace whose “coarse codimension” is 2 can coarsely separate
the ambient space. We must also study vertex rigidity phenomena in the presence
of incident edge spaces of coarse codimension zero and one. In order to make sense
of these different phenomena, for example in order for the coarse codimension of
an edge space in a vertex space to be well-defined, we will add a hypothesis that
the edge spaces satisfy a coarse version of the finite type property. In other words,
the stabilizers of edges (and also of positive depth vertices) are required to be
“coarse finite type groups” (see section 2.6). The coarse finite type hypothesis is
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