Cannon-Thurston maps for Coxeter groups with signature $(n-1,1)$

TL;DR

This paper establishes the existence of Cannon-Thurston maps for certain Coxeter groups with signature (n-1,1), by constructing an isometric action on an ellipsoid and analyzing the limit set of the group.

Contribution

It proves the existence of Cannon-Thurston maps for Coxeter groups with specific signature conditions and constructs an isometric action on an ellipsoid with the Hilbert metric.

Findings

Cannon-Thurston maps exist for Coxeter groups with signature (n-1,1) under certain conditions.

The limit set of the Coxeter group coincides with the accumulation points of roots.

An isometric action on an ellipsoid with the Hilbert metric is constructed.

Abstract

For a Coxeter group $W$ we have an associating bi-linear form $B$ on suitable real vector space. We assume that $B$ has the signature $(n-1,1)$ and all the bi-linear form associating rank $n' (\ge 3)$ Coxeter subgroups generated by subsets of $S$ has the signature $(n',0)$ or $(n'-1,1)$. Under these assumptions, we see that there exists the Cannon-Thurston map for $W$, that is, the $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. To see this we construct an isometric action of $W$ on an ellipsoid with the Hilbert metric. As a consequence, we see that the limit set of $W$ coincides with the set of accumulation points of roots of $W$.