Edge Inversions in $(P_k)$-closed Groups

TL;DR

This paper constructs specific $(P_2)$-closed groups acting on trees where all edge inversions have infinite order, answering a question by Tornier and exploring the range of possible edge inversion orders.

Contribution

It introduces new $(P_2)$-closed groups with infinite and arbitrarily high finite edge inversion orders, addressing a previously open question.

Findings

Existence of $(P_2)$-closed groups with infinite order edge inversions

Construction of groups with arbitrarily high finite edge inversion orders

Negative answer to Tornier's question about edge inversion orders

Abstract

We construct $(P_2)$-closed groups acting on $T_3$ in which all edge inversions have infinite order. This provides a negative answer to a question posed by Tornier. We also construct a family of $(P_2)$-closed groups for which the smallest order of an edge inversion is an arbitrarily high finite number.