We define a generalization of the fixed point set, called the bounded fixed set, for a group acting by isometries on a metric space. An analogue of the P. A. Smith theorem is proved for metric spaces of finite asymptotic dimension, which relates the coarse homology of the bounded fixed set to the coarse homology of the total space.
We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions are finite covers of conjugate actions. As a consequence, we produce continuous families of non-quasi-isometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental groups.
We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar way to classically monotone maps, and prove that coarsely monotone maps are stable under those pullbacks which exist in the coarse category. For the case of maps between proper metric spaces, we exhibit some connections between the coarse and classical notions of monotone and ligh…
We study the geometry of the Thurston metric on the Teichmüller space $\mathcal{T}(S)$ of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus, $S_{1,1}$. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden's theorem.
We introduce a generalization for bounded geometry that we call bounded scale measure. We show that bounded scale measure is a coarse invariant unlike bounded geometry. We then show equivalent definitions for spaces with bounded scale measure and show other properties that spaces with bounded scale measure satisfy. From there, we generalize property A from uniformly discrete metric spaces to large scale spaces with bounded geometry. Lastly, we construct a definition for property A for large scale spaces with bounded scale measure and show that this definition of property A is a coarse invarian…
A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same constants by some given set of leaves. For instance, the following properties are proved. Either all dense leaves without holonomy are equi-coarsely quasi-isometric to each other, or else there exist residually many dense leaves without holonomy such that each of them is coarsely qu…
We prove that each coarsely homogenous separable metric space $X$ is coarsely equivalent to one of the spaces: the sigleton, the Cantor macro-cube or the Baire macro-space. This classification is derived from coarse characterizations of the Cantor macro-cube and of the Baire macro-space given in this paper. Namely, we prove that a separable metric space $X$ is coarsely equivalent to the Baire macro-space if any only if $X$ has asymptotic dimension zero and has unbounded geometry in the sense that for every $δ$ there is $ε$ such that no $ε$-ball in $X$ can be covered by finitely many sets of di…
This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study "Infinite groups as geometric objects", as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can favorably be played for locally compact groups. The development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various me…
Let $Γ$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $Γ$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_Γ$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_Γ$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_Γ$. Consequently, the divergence of $G_Γ$ is linear, or quadratic, or exponential. We also generalize right-angled …
