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The quasi-redirecting (QR) boundary is a close generalization of the Gromov boundary to all finitely generated groups. In this paper, we establish that the QR boundary exists as a topological space for several well-studied classes of…

We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\Omega \subset \mathbb{R}^2$, then there…

We introduce a new quasi-isometry invariant for finitely generated groups and show that every group with this property admits a subshift which is effectively closed by patterns and that cannot be realized as the topological factor of any…

This paper has two goals: to present some new results that are necessary for further study and applications of quasi-linear functionals, and, by combining known and new results, to serve as a convenient single source for anyone interested…

We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. While the respective results for the Hausdorff and upper box-counting dimension follow immediately from their…

We establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and…

For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set…

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$\kappa$-to-one in a natural sense for some $\kappa>0$. For non-amenable graphs, all quasi-isometries are quasi-$\kappa$-to-one for any…

Take two isomorphic convex co-compact co-infinite volume Kleinian groups, whose regular sets are diffeomorphic. The quotient of hyperbolic 3-space by these groups gives two hyperbolic 3-manifolds whose scattering operators may be compared.…

We show that, in dimensions $n\geq 3$, continuity and boundedness do not restore the Sobolev regularity conjecture of Iwaniec and Martin for weakly quasiregular mappings below the critical exponent. For every bounded domain…

We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0)-spaces,…

We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch's approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to…

Let G and F be finitely generated groups with infinitely many ends and let A and B be graph of groups decompositions of F and G such that all edge groups are finite and all vertex groups have at most one end. We show that G and F are…

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using our methods we then prove that a finitely generated group $H$ admitting a quasi-isometric map $\phi$ into a…

We give a necessary and sufficient condition so that a pair of disjoint Jordan regions in the sphere can be quasiconformally mapped to a pair of disks. As a consequence, we obtain a simple characterization that involves Lipschitz functions…

A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost…

We show for very general classes of measures on locally compact second countable groups that every Borel measurable quasimorphism is at bounded distance from a quasi-biharmonic one. This allows us to deduce non-degenerate central limit…

We consider a finite-dimensional, locally finite CAT(0) cube complex X admitting a co-compact properly discontinuous countable group of automorphisms G. We construct a natural compact metric space B(X) on which G acts by homeomorphisms, the…

We study geometric and topological properties of infinite graphs that are quasi-isometric to a planar graph of bounded degree. We prove that every locally finite quasi-transitive graph excluding a minor is quasi-isometric to a planar graph…

In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for…