Detecting Free Group Automorphisms via Virtual Homology Representations

TL;DR

This work develops a homological framework to detect automorphisms of the free group $F_n$ via the End($F_n$)-action on the first homology of finite characteristic covers, establishing that such actions are asymptotically faithful and thus yield asymptotic linearity results. By relating deck group actions to homology, the authors extend the approach to Out($F_n$) and to mapping-class-type questions on surfaces of the same homotopy type as graphs, using towers of covers and virtual homology representations. A central contribution is a Scott-type Abelian-cover lifting theorem, which shows that elevations of loops to iterated Abelian covers can be made simple, enabling a robust disjointness and simplicity detection mechanism via algebraic intersection in covers. The results provide a bridge between graph automorphisms and surface mapping classes through homological data, yielding practical criteria to recognize whether a homotopy equivalence is induced by a homeomorphism and illuminating limitations of nilpotent quotients in detecting epimorphisms.

Abstract

Let $F_n= F\langle x_1,...,x_n\rangle$ denote the free group of rank $n\ge 2$ and let $\mathrm{End}(F_n)$ be the endomorphism monoid of $F_n$. We show that automorphisms of $F_n$ are detected via the $\mathrm{End}(F_n)$-action on the first integral homology of finite characteristic covers of the wedge of $n\ge 2$ circles $R_n$. This gives a homological characterization of homotopy equivalences of $R_n$ that we utilize to show that $\mathrm{End}(F_n)$ is asymptotically linear. We extend these results by showing that the $\mathrm{Out}(F_n)$-action on the homology of iterated covers of a punctured surface $Σ_g^b$ of the same homotopy type as $R_n$ detects homeomorphisms of $Σ_g^b$ in homotopy classes of homotopy equivalences of $R_n$.

Figures (7)

• Figure 1: The mod $2$ homology cover $p\colon X\to R_2$.
• Figure 2: The Whitehead graph $\mathrm{Wh}(\alpha_1,X)$ in Figure a) for the cylically reduced word $\alpha_1 = x_2x_1x_2^{-2}x_1^{-2}$ in the rank two free group $F\langle x_1,x_2\rangle$ and in Figure b) for $\alpha_1 = x_4x_3x_2x_1x_4^{-2}x_3^{-2}x_2^{-2}x_1^{-2}$ in $F\langle x_1,x_2,x_3,x_4\rangle$. Both graphs are connected and admit no cut vertices.
• Figure 3: Band-summing a bounding pair $y,z$ of genus $g'=0$ to a separating curve $w$ of the same genus that lifts to nonseparating curves in the order $2$ cyclic cover. The homologous curves $y,z$ lift to nonhomologous curves $y_1',z_1'$.
• Figure 4: The closed curve $x=\alpha_1^2\beta_2\alpha_2\beta_2^{-1}\alpha_2^{-1}$ defines a multiple of $[\alpha_1]\in H_1(\Sigma,\mathbb{Z})$.
• Figure 5: $z'\cup y'$ defines a simple subloop of $x$.

Theorems & Definitions (40)

• Proposition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of Proposition \ref{['asymptotic_aut']}
• proof : Proof of Theorem \ref{['main_theorem']}