Rationally smooth elements of Coxeter groups and triangle group avoidance

TL;DR

This paper investigates a class of infinite Coxeter groups defined by avoiding certain subgroups, showing rational smoothness can be detected via Poincaré polynomial coefficients and providing enumeration results.

Contribution

It introduces a new family of Coxeter groups characterized by subgroup avoidance and establishes criteria for rational smoothness using Poincaré polynomial coefficients.

Findings

Rationally smooth elements are identifiable by specific Poincaré polynomial coefficients.

A factorization theorem for the Poincaré polynomial of rationally smooth elements is proved.

Many infinite Coxeter groups have finitely many rationally smooth elements, with explicit classifications in special cases.

Abstract

We study a family of infinite-type Coxeter groups defined by the avoidance of certain rank 3 parabolic subgroups. For this family, rationally smooth elements can be detected by looking at only a few coefficients of the Poincar\'{e} polynomial. We also prove a factorization theorem for the Poincar\'{e} polynomial of rationally smooth elements. As an application, we show that a large class of infinite-type Coxeter groups have only finitely many rationally smooth elements. Explicit enumerations and descriptions of these elements are given in special cases.