One important part of geometric group theory is the classification of finitely
generated groups up to quasi-isometry. This paper addresses that question for
many groups which split as graphs of groups. A typical question in this context is:
suppose G splits as a graph of groups and H is quasi-isometric to G, must H split
in a similar way?
The first example of this type is Stallings Ends Theorem, which implies that if
G splits with finite edge groups, then so must H. This result was recently refined in
work of Papasoglu and Whyte [PW02] which implies that if G and H are accessible
groups then they have the same quasi-isometry types of one-ended factors. After
Stallings theorem, the next result of this type came in work of Kapovich and Leeb
[KL97] which shows that if M is a Haken 3-manifold then the splitting of π1(M)
induced by the geometric decomposition of M is preserved by quasi-isometries.
Farb and Mosher [FM99] showed that if G splits as a graph of Z’s with solvable
fundamental group — a solvable Baumslag-Solitar group — then so must H up
to finite groups. Papasoglu [Pap05] showed that if G splits with two-ended edge
groups, then H must as well. Mosher, Sageev and Whyte [MSW03] showed that if
G splits as a “bushy” graph of coarse Poincar´ e duality groups of constant dimension,
then so must H.
Not all splittings exhibit this type of rigidity. For instance, cocompact lattices
in SL(2, R) × SL(2, R) are all quasi-isometric, however, by a theorem of Margulis
the irreducible ones do not split while reducible ones split as amalgams in a myriad
of ways ([Mar81], and see also [MS04]).
In this paper, we focus on graphs of groups whose vertex groups are “manifold-
like”, for example fundamental groups of closed aspherical manifolds, and more
generally finitely presented Poincar´ e duality groups. Since we are dealing with
quasi-isometry issues, it turns out that the natural class of groups to work with are
coarse Poincar´ e duality groups, introduced by Kapovich and Kleiner [KK99]. In a
previous paper [MSW03] we studied a special case in which all of the vertex and
edge groups have the same dimension, equivalently, every edge-to-vertex injection
in the graph of groups has finite index image; such graphs of groups are called
(geometrically) homogeneous. In this paper, we address more general cases.
Papasoglu’s preprint [Pap07] independently studies graphs of groups with a
similar focus and overlapping results to ours; we explain below some of the similar-
ities and differences of our approaches and results.
The main theorems 1.5, 1.6 and 1.7 are somewhat involved, so before launching
into a full discussion of them, we begin with some applications, chosen from among
the gamut of settings to which the main theorems apply.1
1The main theorems, and several of the applications, were announced in [MSW00].