# Realizations of groups of piecewise continuous transformations of the circle

## TL;DR

This paper investigates the realizability of subgroups within the group of piecewise continuous transformations of the circle, establishing conditions under which such subgroups can be lifted to permutations and characterizing their uniqueness.

## Contribution

It proves that all finitely generated abelian subgroups are realizable and identifies specific non-realizable subgroups, also characterizing the unique realizations of interval exchange groups.

## Key findings

- Finitely generated abelian subgroups are realizable.
- Certain subgroups of interval exchanges are not realizable.
- Interval exchange groups have exactly two realizations.

## Abstract

We study the near action of the group PC of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of PC is said to be realizable if it can be lifted to a group of permutations of the circle. We show that every finitely generated abelian subgroup of PC is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.