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 $r\geq1$ and that the canonical map $\St_r(R)\to\GL_r(R)$ is an isomorphism for every $r\geq3$. The homology calculation combines simultaneous extensions of ordered frames with scalar actions of the multiplicative groups of finite fields on their stabilizers. The presentation associated with the same frame complex defines a surjective section of the Steinberg map. An explicit finite presentation of $R^\times$ then follows from the theorem of Krsti\'c and McCool. We formulate separate criteria for acyclicity and for the Steinberg comparison over other rings.