General linear and Steinberg groups over the Leavitt algebra $L_{{\mathbb{F}}_2}(1,2)$

Abstract

Let $R=L_{\F_2}(1,2)$. We prove that $\GL_r(R)$ is integrally acyclic for every $r\ge 1$ and that the canonical map $\St_r(R)\to\GL_r(R)$ is an isomorphism for every $r\ge 3$. The unit group $R^{\times }$ is finitely presented, and we describe an explicit finite presentation. The homology calculation uses leaf coordinates, simultaneous extensions of ordered frames, and finite-field actions on stabilizers. The Steinberg argument lifts relations from a simply connected frame complex and refines coordinates. We also state separate criteria for the two arguments over rings of characteristic two.