Twisted conjugacy in $\mathrm{SL}_{n}$ and $\mathrm{GL}_{n}$ over subrings of $\overline{\mathbb{F_p}}(t)$

Abstract

Let \varphi\colon G\to G be an automorphism of an infinite group G . One has an equivalence relation \sim_{\varphi} on G defined as x\sim_{\varphi} y if there exists a z\in G such that y=zx\varphi(z^{-1}) . The equivalence classes are called \varphi -twisted conjugacy classes, and the set G/{\sim}_{\varphi} of equivalence classes is denoted by \mathcal{R}(\varphi) . The cardinality R(\varphi) of \mathcal{R}(\varphi) is called the Reidemeister number of \varphi . We write R(\varphi)=\infty when \mathcal{R}(\varphi) is infinite. We say that G has the R_{\infty} -property if R(\varphi)=\infty for every automorphism \varphi of G . We show that the groups G=\mathrm{GL}_{n}(R), \mathrm{SL}_{n}(R) have the R_{\infty} -property for all n\ge 3 when F[t]\subset R\subsetneq F(t) , where F is a subfield of \overline{\mathbb{F_p}} . When n\ge 4 , we show that any subgroup H\subset \mathrm{GL}_{n}(R) that contains \mathrm{SL}_{n}(R) also has the R_{\infty} -property.