A Super Version of a Theorem of Fricke-Klein

TL;DR

This paper explores the supergeometric character variety of the supergroup OSp(1|2), extending classical results by Fricke and Klein to the supergroup context, with initial focus on the free group on two generators.

Contribution

It provides the first algebraic description of the character variety for OSp(1|2), a supergroup, as a supergeometric analogue of the classical Fricke-Klein theorem.

Findings

Explicit description of the invariant ring for OSp(1|2)

Extension of classical character variety results to supergroups

Initial insights into the character stack for OSp(1|2)

Abstract

We start studying the character variety of the algebraic supergroup OSp(1|2) from the algebraic perspective. We do this by first investigating the specific case of the character variety of the free group on two letters and try to describe the ring of invariants with respect to the conjugation action. The explicit description of the corresponding character variety for SL(2) was done by Fricke and Klein, so this can be seen as a variant of this theorem for its supergeometric counterpart OSp(1|2) and briefly touch upon the character stack for OSp(1|2)