An endomorphism $\phi: G\to G$ of a group yields an action of $G$ on itself, known as the $\phi$-twisted conjugacy action, where $(g,x)\mapsto gx\phi(g^{-1})$. The group $G$ is said to have the property $R_\infty$ if, for any automorphism $\phi$ of $G$, the orbit space of the $\phi$-twisted conjugacy action is infinite. This notion, and the related notion of Reidemeister number, originated from Nielsen fixed point theory.

It is an interesting problem to decide, given an infinite group $G$ whether or not $G$ has property $R_\infty$. We will consider the problem in the case when $G=GL(n,R), SL(n,R), n\ge 2$, when $R$ is either a polynomial ring or a Laurent polynomial ring over a finite field $\mathbb{F}_q$.

The talk is based on recent joint work with Oorna Mitra.

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Last updated: 15 Jul 2024