1. Introduction and statement of results. The geometry of discrete metric spaces has recently been a very active research area. The motivation comes primarily from two (not completely separate) areas of mathematics. The first is the study of noncompact manifolds equipped with metrics well defined up to uniformly bounded distortion. The equivalence classes of these manifolds arise naturally in the study of compact manifolds-for example, as universal covers or leaves of foliations. To focus attention purely on the large-scale structure, the local topology is thrown away by passing to a discrete subset, called a net, which evenly fills out the space. A typical example is Z n in R n . Two nets in a space are bilipschitz equivalent on the large scale, but not the small. This leads to the notion of a quasi-isometry of metric spaces and gives a functor from manifolds of bounded geometry, up to bounded distortion, to discrete metric spaces, up to quasi-isometry.The second source of motivation and examples is geometric group theory. A finitely generated group can be given a metric by measuring the distance from an element to the identity by the minimal length of a word in the generators representing the element. This extends to a metric by left translation invariance. This metric, called the (left) word metric, is well defined up to bilipschitz equivalence. (This can be seen simply by expanding one set of generators in another.) One can then study the group via the geometry of this space.The coarse geometry of manifolds and geometric group theory are related, most obviously by the fundamental group. If M is a compact manifold, thenM has obvious nets given by the orbits of the action of π 1 (M). These nets are canonically identified, at least as sets, with π 1 (M). Conversely, if is any finitely presented group, there is a compact manifold with π 1 = . It is a fundamental observation in the subject that the quasi-isometry type of as a net agrees with the word metric. More generally, the Milnor-Svarc theorem says that any path metric space on which acts cocompactly by isometries is quasi-isometric to with its word metric. Thus, for instance, any two compact hyperbolic manifolds of the same dimension have quasi-isometric fundamental groups. The same holds for other locally symmetric spaces. This is the sense in which we expect the word metric to capture the "geometric type" of a group.The reader may have noticed a small distinction between the geometric and algebraic viewpoints: nets are well defined only up to quasi-isometry, while the word met-

Given a bounded valence, bushy tree T , we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T . This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan's theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first examples of two finitely generated groups which are quasi-isometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries.

In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely generated group. This paper is the first in a sequence of papers proving results announced in our 2007 article "Quasi-isometries and rigidity of solvable groups." In particular, this paper contains many steps in the proofs of quasi-isometric rigidity of lattices in Sol and of the quasi-isometry classification of lamplighter groups. The proofs of those results are completed in "Coarse differentiation of quasi-isometries II; Rigidity for lattices in Sol and Lamplighter groups." The method used here is based on the idea of coarse differentiation introduced in our 2007 article.

Abstract. We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

Let G, F be finitely generated groups with infinitely many ends and let π1(Γ, A), π1(∆, B) be graph of groups decompositions of F, G such that all edge groups are finite and all vertex groups have at most one end. We show that G, F are quasi-isometric if and only if every one-ended vertex group of π1(Γ, A) is quasi-isometric to some one-ended vertex group of π1(∆, B) and every one-ended vertex group of π1(∆, B) is quasi-isometric to some one-ended vertex group of π1(Γ, A). From our proof it also follows that if G is any finitely generated group, of order at least three, the groups: G * G, G * Z, G * G * G and G * Z/2Z are all quasi-isometric.