Poincare profiles are a family of analytically defined coarse invariants, which can be used as obstructions to the existence of coarse embeddings between metric spaces. In this paper we calculate the Poincare profiles of all connected unimodular Lie
groups, Baumslag-Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. In the case of Lie groups, we obtain a dichotomy which extends both the dichotomy separating rank one and higher rank semisimple Lie groups and the dichotomy separating connected solvable unimodular Lie groups of polynomial and exponential growth. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. Our results have many consequences for coarse embeddings, for instance we deduce that for groups of the form $Ntimes S$, where $N$ is a connected nilpotent Lie group, and $S$ is a simple Lie group of real rank 1, both the growth exponent of $N$, and the Ahlfors-regular conformal dimension of $S$ are non-decreasing under coarse embeddings. These results are new even in the quasi-isometric setting and give obstructions to quasi-isometric embeddings which in many cases are stronger than those previously obtained by Buyalo-Schroeder.
We define Poincar{e} profiles of Dirichlet type for graphs of bounded degree, in analogy with the Poincar{e} profiles (of Neumann type) defined in [HMT19]. The obvious first definition yields nothing of interest, but an alternative definition yields
a spectrum of profiles which are quasi-isometry invariants and monotone with respect to subgroup inclusion. Moreover, in the extremal cases $p=1$ and $p=infty$, they detect the Fo lner function and the growth function respectively.
We investigate groups whose Cayley graphs have poor-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Timar if and only if it is virtually free. We then prove a gap theorem for
connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type $F$ with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.
We generalize Gruber--Sistos construction of the coned--off graph of a small cancellation group to build a partially ordered set $mathcal{TC}$ of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber
--Sisto coned--off graph. In almost all cases $mathcal{TC}$ is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions $[Gcurvearrowright X] preceq [Gcurvearrowright Y]$ in this poset, there is an embeddeding $iota:P(omega)tomathcal{TC}$ such that $iota(emptyset)=[Gcurvearrowright X]$ and $iota(mathbb N)=[Gcurvearrowright Y]$. We use this poset to prove that there are uncountably many quasi--isometry classes of finitely generated group which admit two cobounded acylindrical actions on hyperbolic spaces such that there is no action on a hyperbolic space which is larger than both.
We explore geometric conditions which ensure a given element of a finitely generated group is, or fails to be, generalized loxodromic; as part of this we prove a generalization of Sistos result that every generalized loxodromic element is Morse. We p
rovide a sufficient geometric condition for an element of a small cancellation group to be generalized loxodromic in terms of the defining relations and provide a number of constructions which prove that this condition is sharp.
We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincar{e} profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Tim{a}r. In
this paper we focus on properties of the Poincar{e} profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these profiles and conformal dimension. As applications, we use these invariants to show the non-existence of coarse embeddings in a variety of examples.
We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is admitting exponentially many fat bigons, a
nd it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the former cannot be coarsely embedded into the latter. Other examples include certain lacunary hyperbolic and certain small cancellation groups.
We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any finite col
lection of finitely generated groups $mathcal{H}$ each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi--isometry types of one--ended groups which are hyperbolic relative to $mathcal{H}$. The groups are constructed using small cancellation theory over free products.
Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalize these strategies by viewing any geodesic metric space as a countable
union of stable subspaces: we show that every stable subgroup is a quasi--convex subset of a set in this collection and that the Morse boundary is recovered as the direct limit of the usual Gromov boundaries of these hyperbolic subspaces. We use this approach, together with results of Leininger--Schleimer, to deduce that there is no purely geometric obstruction to the existence of a non-virtually--free convex cocompact subgroup of a mapping class group. In addition, we define two new quasi--isometry invariant notions of dimension: the stable dimension, which measures the maximal asymptotic dimension of a stable subset; and the Morse capacity dimension, which naturally generalises Buyalos capacity dimension for boundaries of hyperbolic spaces. We prove that every stable subset of a right--angled Artin group is quasi--isometric to a tree; and that the stable dimension of a mapping class group is bounded from above by a multiple of the complexity of the surface. In the case of relatively hyperbolic groups we show that finite stable dimension is inherited from peripheral subgroups. Finally, we show that all classical small cancellation groups and certain Gromov monster groups have stable dimension at most 2.
