## Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1. (Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.)(French)Zbl 0678.53042

A continuous map $$f$$ between metric spaces $$X_1$$ and $$X_2$$ is a quasiisometry if there are positive constants $$C$$ and $$L$$ such that $$-C+(1/L)d(x,y)\le d(fx,fy)\le C+Ld(x,y)$$ for all points $$x,y$$ in $$X_ 1$$. Now let $$\tilde M$$ denote a symmetric space of rank 1 (i.e. strictly negative sectional curvature) and let $$\tilde M(\infty)$$ denote the boundary sphere of $$\tilde M$$. Morse showed essentially that any quasiisometry $$f: \tilde M\to \tilde M$$ extends to a continuous map $$\bar f: \tilde M(\infty)\to \tilde M(\infty)$$. Mostow and the author showed that $$\bar f$$ is in fact quasiconformal with respect to a non-Riemannian Carnot-Carathéodory metric $$\delta$$ on $$\tilde M(\infty)$$. The metric $$\delta$$ actually depends on the choice of a point $$x$$ in $$\tilde M(\infty)$$ and a horosphere $$H$$ at $$x$$, but quasiconformality and 1-quasiconformality for a function $$f: \tilde M(\infty)\to \tilde M(\infty)$$ are independent of $$x$$ and $$H$$.
In this paper the author proves the following main theorem. Let $$\tilde M$$ be the $$n$$-dimensional quaternionic hyperbolic space or the Cayley plane, and let $$f: \tilde M\to \tilde M$$ be a quasiisometry. Then there is an isometry $$g: \tilde M\to \tilde M$$ such that $$d(fp,gp)\le C$$ for some $$C>0$$ and all points $$p$$ in $$\tilde M$$.
We sketch a proof. Given a point $$x$$ in $$\tilde M(\infty)$$ one may identify $$\tilde M(\infty)-\{x\}$$ with a simply connected 2-step nilpotent group $$N$$ of isometries of $$\tilde M$$ that acts simply transitively on horospheres at $$x$$. The point $$x$$ also determines a derivation $$\alpha$$ of the Lie algebra of $$N$$. If $$f: \tilde M\to \tilde M$$ is a quasiisometry, then for almost all points $$n$$ of $$N$$ the extension $$\bar f: \tilde M(\infty)\to \tilde M(\infty)$$ induces a “$$\delta$$-differential” $$df_n: N\to N$$, which is an automorphism of $$N$$ that commutes with $$\alpha$$. If $$\tilde M$$ is a quaternionic hyperbolic space or the Cayley plane, then the centralizer of $$\alpha$$ in $$\operatorname{Aut}(N)$$ is the product of $$\{e^{t\alpha}: \alpha \in\mathbb{R}\}$$ with a compact group arising from isometries of $$M$$. (This fact is not true for the real and complex hyperbolic spaces, and in fact the main theorem is not true for these spaces). From this it follows that the extension $$\bar f: \tilde M(\infty)\to \tilde M(\infty)$$ is 1-quasiconformal with respect to the metric $$\delta$$ mentioned above. An argument of Mostow shows that any 1-quasiconformal map of $$\tilde M(\infty)$$ is induced by an isometry of $$\tilde M$$, and the main theorem now follows.

### MathOverflow Questions:

Quasi-isometry groups of metric spaces

### MSC:

 53C35 Differential geometry of symmetric spaces 53C20 Global Riemannian geometry, including pinching 22E40 Discrete subgroups of Lie groups
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