Distribution of accumulation points of roots for type $(n-1,1)$ Coxeter groups

TL;DR

This paper studies the accumulation points of normalized roots in infinite Coxeter groups of type (n-1,1), proving a conjecture and characterizing these points as the closure of an orbit, with insights into fixed points of the group action.

Contribution

It proves a conjecture regarding accumulation points for type (n-1,1) Coxeter groups and characterizes these points as the closure of a specific orbit.

Findings

Confirmed the conjecture for type (n-1,1) Coxeter matrices.

Characterized accumulation points as the closure of an orbit.

Analyzed properties of fixed points in the group action.

Abstract

In this paper, we investigate the set of accumulation points of normalized roots of infinite Coxeter groups for certain class of their action. Concretely, we prove the conjecture proposed in [6, Section 3.2] in the case where the equipped Coxeter matrices are of type $(n-1,1)$, where $n$ is the rank. Moreover, we obtain that the set of such accumulation points coincides with the closure of the orbit of one point of normalized limit roots. In addition, in order to prove our main results, we also investigate some properties on fixed points of the action.