Asymptotic cones and boundaries of CAT(0) groups

TL;DR

This paper investigates the relationship between asymptotic cones and the Tits boundary of proper cocompact CAT(0) spaces, revealing how these structures are interconnected through limits and topologies.

Contribution

It establishes canonical connecting maps between asymptotic cones and shows the direct limit is isometric to the Euclidean cone on the Tits boundary, linking visual and Tits topologies.

Findings

Asymptotic cones admit canonical connecting maps.

The direct limit of asymptotic cones is isometric to the Euclidean cone on the Tits boundary.

Maps between asymptotic cones induce maps between Tits boundaries.

Abstract

It is well known that the Tits boundary of a proper cocompact CAT(0) space embeds into every asymptotic cone of the space. We explore the relationships between the asymptotic cones of a CAT(0) space and its boundary under both the standard visual (i.e. cone) topology and the Tits metric. We show that the set of asymptotic cones of a proper cocompact CAT(0) space admits canonical connecting maps under which the direct limit is isometric to the Euclidean cone on the Tits boundary. The resulting projection from any asymptotic cone to the Tits boundary is determined by the visual topology; on the other hand, the visual topology can be recovered from the connecting maps between asymptotic cones. We also demonstrate how maps between asymptotic cones induce maps between Tits boundaries.