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Register a new userSublinear quasiconformality and the large-scale geometry of Heintze groups
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Gabriel Pallier
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This article analyzes sublinearly quasisymmetric homeo-morphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and spaces. It is proven that those homeomorphisms lack analytical properties but preserve a conformal dimension and appropriate function spaces, distinguishing certain (nonsymmetric) Riemannian negatively curved homogeneous spaces, and Fuchsian buildings, up to sublinearly biLipschitz equivalence (generalized quasiisometry).
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Large-scale sublinearly Lipschitz maps have been introduced by Yves Cornulier in order to precisely state his theorems about asymptotic cones of Lie groups. In particular, Sublinearly biLipschitz Equivalences (SBE) are a weak variant of quasiisometries, with the only requirement of still inducing biLipschitz maps at the level of asymptotic cones. We focus here on hyperbolic metric spaces and study properties of their boundary extensions, reminiscent of quasiM{o}bius mappings. We give a dimensional invariant on the boundary that allows to distinguish hyperbolic symmetric spaces up to SBE, answering a question of Druc{t}u.
We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S); a characterization of the image of the curve-complex projection map from MCG(S) to the product of the curve complexes of essential subsurfaces of S; and a construction of Sigma-hulls in MCG(S), an analogue of convex hulls.
We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, infty)rightarrowmathtt{Aut}(G)$, $lambdamapstodelta_lambda$, so that $ d(delta_lambda x,delta_lambda y) = lambda d(x,y)$, for all $x,yin G$ and all $lambda>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is biLipschitz equivalent to one that admits a one-parameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in $[1,infty)$. Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. Finally, we characterize metric Lie groups admitting a one-parameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors. Such metric spaces appear as tangents of doubling metric spaces with unique tangents.
In this paper, we construct Holder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $mathbb{H}$. Pansu and Gromov observed that any surface embedded in $mathbb{H}$ has Hausdorff dimension at least 3, so there is no $alpha$-Holder embedding of a surface into $mathbb{H}$ when $alpha>frac{2}{3}$. Zust improved this result to show that when $alpha>frac{2}{3}$, any $alpha$-Holder map from a simply-connected Riemannian manifold to $mathbb{H}$ factors through a metric tree. In the present paper, we show that Zusts result is sharp by constructing $(frac{2}{3}-epsilon)$-Holder maps from $D^2$ and $D^3$ to $mathbb{H}$ that do not factor through a tree. We use these to show that if $0<alpha < frac{2}{3}$, then the set of $alpha$-Holder maps from a compact metric space to $mathbb{H}$ is dense in the set of continuous maps and to construct proper degree-1 maps from $mathbb{R}^3$ to $mathbb{H}$ with Holder exponents arbitrarily close to $frac{2}{3}$.
The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given class of regularity. It has been adapted to several settings, among which the one of Carnot groups. However, the target space has generally been assumed to be equal to R^d for some d $ge$ 1. We focus here on the extendability problem for general ordered pairs (G_1,G_2) (with G_2 non-Abelian). We analyze in particular the case G_1 = R and characterize the groups G_2 for which the Whitney extension property holds, in terms of a newly introduced notion that we call pliability. Pliability happens to be related to rigidity as defined by Bryant an Hsu. We exploit this relation in order to provide examples of non-pliable Carnot groups, that is, Carnot groups so that the Whitney extension property does not hold. We use geometric control theory results on the accessibility of control affine systems in order to test the pliability of a Carnot group. In particular, we recover some recent results by Le Donne, Speight and Zimmermann about Lusin approximation in Carnot groups of step 2 and Whitney extension in Heisenberg groups. We extend such results to all pliable Carnot groups, and we show that the latter may be of arbitrarily large step.
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