Examples of Non-Rigid CAT(0) Groups from the Category of Knot Groups

TL;DR

This paper demonstrates that the knot group of any connected sum of two non-trivial torus knots has uncountably many distinct CAT(0) boundaries, expanding understanding of boundary diversity in CAT(0) groups.

Contribution

It extends the class of known CAT(0) groups with uncountably many boundaries by focusing on knot groups of connected sums of torus knots.

Findings

Knot groups of connected sums of two non-trivial torus knots have uncountably many CAT(0) boundaries.

The result generalizes previous examples of CAT(0) groups with multiple boundaries.

Supports the idea that boundary complexity is common among certain knot groups.

Abstract

C Croke and B Kleiner have constructed an example of a CAT(0) group with more than one visual boundary. J Wilson has proven that this same group has uncountably many distinct boundaries. In this article we prove that the knot group of any connected sum of two non-trivial torus knots also has uncountably many distinct CAT(0) boundaries.