We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\mathbb{H}^n$, $n>1$, is close to some isometry up to proximity order $\sqrt{\varepsilon}+\varepsilon$ in the uniform norm, and up to proximity order $\varepsilon$ in the $L_p^1$-norm. We give examples showing the asymptotic sharpness of our results.
In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].
In this paper, we continue with the results in \cite{Pg} and compute the group of quasi-isometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member of the subclass has to be polycyclic, and is virtually a lattice in an abelian-by-abelian solvable Lie group. We also give an example of a unimodular solvable Lie group that is not quasi-isometric to any finitely generated group, as well deduce some quasi-isometric rigidity results.
Let $(X,d,\mathfrak{m})$ be a metric measure space. The study of the Wasserstein space $(\mathbb{P}_p(X),\mathbb{W}_p)$ associated to $X$ has proved useful in describing several geometrical properties of $X.$ In this paper we focus on the study of isometries of $\mathbb{P}_p(X)$ for $p \in (1,\infty)$ under the assumption that there is some characterization of optimal maps between measures, the so called Good transport behaviour $GTB_p$. Our first result states that the set of Dirac deltas is invariant under isometries of the Wasserstein space. Additionally we obtain that the isometry groups o…
In this paper we provide the final steps in the proof, announced by Eskin-Fisher-Whyte, of quasi-isometric rigidity of a class of non-nilpotent polycyclic groups. To this end, we prove a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces and combine it with work of Eskin-Fisher-Whyte and Peng on the structure of quasi-isometries of certain solvable Lie groups.
We prove the following rank rigidity result for proper CAT(0) spaces with one-dimensional Tits boundaries: Let $Γ$ be a group acting properly discontinuously, cocompactly, and by isometries on such a space $X$. If the Tits diameter of $\partial X$ equals $π$ and $Γ$ does not act minimally on $\partial X$, then $\partial X$ is a spherical building or a spherical join. If $X$ is also geodesically complete, then $X$ is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of $\partial X$, doe…
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 [EFW0]. 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 res…
In this paper, we first prove the optimal lower bound for Alexandrov angle rigidity of torsion elliptic isometries on any complete CAT($κ$) space, which, when attained, leads to an embedded 2-flat in the tangent cone invariant under the induced action of the isometry. Next, we will prove similar result for action of symmetry groups of either a regular orthoplex, a regular hypercube, a regular dodecahedron or a regular icosahedron on a set of points in any complete CAT($κ$) space in a way corresponds to the set of vertices of the polytope, the angle made at the circumcenter by any pair of point…
Torsion subgroups and fixed-point rigidity in CAT(0) geometry
Qiuzhen College, Tsinghua University（清华大学邱班）；Keio University（庆应义塾大学）；Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学多学科研究院）；University of Geneva（日内瓦大学）；Uppsala University（乌普萨拉大学）；Tsinghua University（清华大学）；BIMSA（北京国际数学研究中心）
We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a space as above has some negative curvature behaviour then it cannot support a thin action: this generalizes the classical Margulis Lemma to a broader class of spaces.
We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers. Secondly, we characterise geometric properties of a group (coarse e…
