Signature for piecewise continuous groups

Abstract

Let $\widehat{{\mathrm{PC}}^{\bowtie }}$ be the group of bijections from $[0,1[$ to itself which are continuous outside a finite set. Let ${\mathrm{PC}}^{\bowtie }$ be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of ${\mathrm{PC}}^{\bowtie }$ vanishes. That is, the quotient map $\widehat{{\mathrm{PC}}^{\bowtie }}\to {\mathrm{PC}}^{\bowtie }$ splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, from $\widehat{{\mathrm{PC}}^{\bowtie }}$ to $\mathbb{Z}/2\mathbb{Z}$. Then we use this signature to list normal subgroups of every subgroup $\hat{G}$ of $\widehat{{\mathrm{PC}}^{\bowtie }}$ which contains ${\mathfrak{S}}_{\mathrm{fin}}$ such that $G$, the projection of $\hat{G}$ in ${\mathrm{PC}}^{\bowtie }$, is simple.