Flows and joins of metric spaces

Abstract

We introduce the functor $\ast$ which assigns to every metric space $X$ its symmetric join $\ast X$. As a set, $\ast X$ is a union of intervals connecting ordered pairs of points in $X$. Topologically, $\ast X$ is a natural quotient of the usual join of $X$ with itself. We define an $Isom\left(X\right)$–invariant metric $d_{\ast }$ on $\ast X$.

Classical concepts known for ${ℍ}^n$ and negatively curved manifolds are defined in a precise way for any hyperbolic complex $X$, for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification $\bar{X}=X\bigsqcup \partial X$. They are continuous, $Isom\left(X\right)$–invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry $g\in Isom\left(X\right)$.

For any hyperbolic complex $X$, the symmetric join $\ast \bar{X}$ of $\bar{X}$ and the (generalized) metric $d_{\ast }$ on it are defined. The geodesic flow space $\mathcal{ℱ}\left(X\right)$ arises as a part of $\ast \bar{X}$. $\left(\mathcal{ℱ}\left(X\right),d_{\ast }\right)$ is an analogue of (the total space of) the unit tangent bundle on a simply connected negatively curved manifold. This flow space is defined for any hyperbolic complex $X$ and has sharp properties. We also give a construction of the asymmetric join $X\ast Y$ of two metric spaces.

These concepts are canonical, ie functorial in $X$, and involve no “quasi"-language. Applications and relation to the Borel conjecture and others are discussed.