Periodic families in the homology of $GL_n(F_2)$

TL;DR

This paper constructs infinite families of nonzero homology classes in $GL_n(F_2)$, demonstrating the optimality of the known slope $rac{2}{3}$-stability and extending stability results to related groups using a new Hopf algebra perspective.

Contribution

It introduces a new method to construct homology classes along specific lines, confirming the optimality of the slope $rac{2}{3}$-stability for these groups and related automorphism groups.

Findings

Constructed infinite families of nonzero classes in $H_d(GL_n(F_2))$

Confirmed the optimality of the slope $rac{2}{3}$-stability

Extended stability results to $GL_n(Z)$ and $Aut(F_n)$

Abstract

We construct infinite families of nonzero classes in $H_d(GL_n(F_2);F_2)$ along lines of the form $d =\frac{2}{3}n +$(constant), thereby showing that the known slope $\frac{2}{3}$-stability for these homology groups are optimal. Using the new stability Hopf algebra perspective of Randal-Williams, our computations in addition recover the slope-$\frac{2}{3}$ stability for $GL_n(Z)$ with coefficients in $F_2$, improve that for $Aut(F_n)$ to $\frac{2}{3}$, and demonstrate that those slopes are optimal. Perhaps of independent interest, we also provide a manual for computing stability Hopf algebras over $F_2$.