Strongly contracting geodesics in a tree of spaces (Q2154717)

The celebrated combination theorem of \textit{M. Bestvina} and \textit{M. Feighn} [J. Differ. Geom. 35, No. 1, 85--102 (1992; Zbl 0724.57029)] gives sufficient conditions on a tree of \(\delta\)-hyperbolic metric spaces to be itself \(\delta\)-hyperbolic. It has become a prototype for many theorems which bootstrap properties of vertex and edge spaces into properties of the resulting tree of spaces. Going in a somewhat different direction, one might ask to what extent properties of a vertex space survive in a tree of spaces; this is the direction of the main theorem of this paper. The authors prove that if edge spaces are \textit{strongly contracting} and sufficiently separated in vertex spaces, then strongly contracting geodesics in vertex spaces remain quasiconvex in the tree of spaces. Let \(Y\) be a closed subspace of a proper metric space \(X\) and \(x \in X\). The closest-point projection \(\pi_Y(x)\) of \(x\) to \(Y\) is the set of those \(y \in Y\) realizing \(d(x,Y)\). We say that \(Y\) is \textit{strongly contracting} if there exists \(\sigma \ge 0\) such that for all \(x \in X\), whenever \(x' \in X\) satisfies that \[ \text{if }d(x,x') \le d(x,Y),\text{ then }\operatorname{diam}(\pi_Y(x)\cup\pi_Y(x')) \le \sigma. \] For example, if \(Y\) is a closed subtree (say a geodesic) of a tree \(X\), then \(Y\) is strongly contracting; we may even take \(\sigma =0\) in this case. More generally, geodesics in \(\delta\)-hyperbolic spaces are strongly contracting. In fact, in the same setting above, the authors prove a stronger statement about \((N,0)\)-quasigeodesics in vertex spaces for \(N \ge 1\). This latter statement implies another hyperbolic combination theorem: if every vertex space is proper, geodesic and \(\delta\)-hyperbolic (for some uniform \(\delta\)), every edge space is (uniformly) strongly contracting and edge spaces are (uniformly) sufficiently separated in vertex spaces, then the tree of spaces is itself \(\delta\)-hyperbolic and the vertex spaces are quasiconvex in the tree of spaces. For an example of this latter theorem, one might imagine doubling a word-hyperbolic group \(G\) along a quasiconvex subgroup \(H\), forming the amalgam \(G*_HG\). Since \(G\) is hyperbolic relative to \(H\), by ``cusping off'' the cosets of \(H\) in some Cayley graph of \(G\) and then truncating at a sufficiently high stage, there is a geometric action of \(G\) on a \(\delta\)-hyperbolic space in which orbits stabilized by conjugates of \(H\) (i.e.~the edge spaces in our eventual tree of spaces) are sufficiently separated. They are strongly contracting by general theory, so the theorem applies to show that \(G*_HG\) acts geometrically on a hyperbolic tree of spaces. It was not clear to me in the statement of the theorem above the extent to which one might quantify explicitly the separation required, which might have made the theorem easier to apply in particular examples. The proof of the theorem does not go via the Bestvina-Feighn combination theorem, and the example above shows that it is aiming at a slightly different situation than theirs. In the Bestvina-Feighn paper [loc. cit.] one might want, for example, the fiber subgroup \(G\) in an extension \(G\rtimes \mathbb{Z}\), where quasiconvexity is much too strong to hope for.