We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddi
ngs between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.
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 study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal valu
e of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without $2$-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.
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 introduce a model for random groups in varieties of $n$-periodic groups as $n$-periodic quotients of triangular random groups. We show that for an explicit $d_{mathrm{crit}}in(1/3,1/2)$, for densities $din(1/3,d_{mathrm{crit}})$ and for $n$ large
enough, the model produces emph{infinite} $n$-periodic groups. As an application, we obtain, for every fixed large enough $n$, for every $pin (1,infty)$ an infinite $n$-periodic group with fixed points for all isometric actions on $L^p$-spaces. Our main contribution is to show that certain random triangular groups are uniformly acylindrically hyperbolic.
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.
The contracting boundary of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting bound
ary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.
We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2epsilon)^{1/2p}$-uniformly Lipschitz) with $p$ varying in an
interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal $p$ for which $L^p$-fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every $p_0 in [2, infty)$ for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on $L^p$-spaces that are $(2-2epsilon)^{1/2p}$-uniformly Lipschitz, and this for every $pin [2,p_0]$. To accomplish these goals we find new bounds on the first eigenvalue of the p-Laplacian on random graphs, using methods adapted from Kahn and Szemeredis approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to $L^p$-spaces previous results for Kazhdans Property (T) established by Zuk and Ballmann-Swiatkowski.
We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like l^K
in the length l of the relators, then a.a.s. such a random group has conformal dimension 2+K+o(1). In Gromovs density model, a random group at density d<1/8 a.a.s. has conformal dimension $asymp dl / |log d|$. The upper bound for C(1/8) groups has two main ingredients: $ell_p$-cohomology (following Bourdon-Kleiner), and walls in the Cayley complex (building on Wise and Ollivier-Wise). To find lower bounds we refine the methods of [Mackay, 2012] to create larger `round trees in the Cayley complex of such groups. As a corollary, in the density model at d<1/8, the density d is determined, up to a power, by the conformal dimension of the boundary and the Euler characteristic of the group.
We study a random group G in the Gromov density model and its Cayley complex X. For density < 5/24 we define walls in X that give rise to a nontrivial action of G on a CAT(0) cube complex. This extends a result of Ollivier and Wise, whose walls could
be used only for density < 1/5. The strategy employed might be potentially extended in future to all densities < 1/4.
