Quasi-circles through prescribed points

TL;DR

This paper proves that in certain metric spaces, any finite set of points can be contained within a single quasi-circle, with implications for hyperbolic groups and their geometric properties.

Contribution

It establishes conditions under which points in specific metric spaces lie on a quasi-circle, extending understanding of geometric embeddings in hyperbolic groups.

Findings

Any n points in the space lie on a K-quasi-circle.

Hyperbolic groups without splittings over virtually cyclic subgroups contain geodesic lines in quasi-isometric hyperbolic planes.

Abstract

We show that in an L-annularly linearly connected, N-doubling, complete metric space, any n points lie on a K-quasi-circle, where K depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split over any virtually cyclic subgroup, then any geodesic line in G lies in a quasi-isometrically embedded copy of the hyperbolic plane.