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Realizations of groups of piecewise continuous transformations of the circle

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  • We study the near action of the group $ \mathrm{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 $ \mathrm{PC} $ is said to be realizable if it can be lifted to a group of permutations of the circle.

    We prove that every finitely generated abelian subgroup of $ \mathrm{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.

    Mathematics Subject Classification: Primary: 37E05; Secondary: 20B27, 20F65, 22F05, 37C85.


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  • Figure 1.  Examples of graphs of elements of $ \mathrm{PC}^ {\, \mathrm{\bowtie}}( \mathbf{S}) $ (parameterizing the circle as an interval). The first belongs to $ \mathrm{IET}^+ $; the second belongs to $ \mathrm{IET}^- $; the third belongs to $ \mathrm{IET}^\bowtie \backslash\mathrm{IET}^\pm $. The fourth is a more "typical" element of $ \mathrm{PC}^ {\, \mathrm{\bowtie}}( \mathbf{S}) $. The value at breakpoints is not prescribed, as we consider group elements as defined up to finite indeterminacy

    Figure 2.  Graphs of a 132-flip and a triple flip: in each case there are two hyper-clean lifts, choosing either the endpoints denoted as circles or dots

    Figure 3.  Graphs of $ u $, $ v $, $ w $ and $ s $

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