On Nielsen equivalence classes of two-elements generators of mapping class groups

TL;DR

The paper proves that for every genus $g \ge 8$, the mapping class group $\mathcal{M}_g$ has infinitely many Nielsen equivalence classes of generating pairs, with the same holding for $\mathrm{Sp}(2g,\mathbb{Z})$ via a surjection from $\mathcal{M}_g$. It constructs explicit two-generator pairs $(h_0,\rho_n)$ and shows they generate $\mathcal{M}_g$, then distinguishes their Nielsen classes by the spectrum of the commutator action on $H_1(\Sigma_g;\mathbb{R})$, yielding infinitely many classes; a similar approach yields non-T-equivalence across different $n$. The paper also proves that $\mathrm{SL}(2,\mathbb{Z})$ has only finitely many Nielsen equivalence classes of generating pairs via an amalgamated-product decomposition, and it shows that T-equivalence remains rich for $g \ge 8$ through automorphism considerations. Overall, the results reveal a highly intricate Nielsen-equivalence landscape for mapping class groups and their symplectic images, with explicit generating pairs and spectral invariants driving the distinctions.

Abstract

We show that there are infinitely many Nielsen equivalence classes of the mapping class group of a closed oriented surface of genus at least eight.

Figures (4)

• Figure 1: Simple closed curves $a,b,c,d,x,y,z$.
• Figure 2: The rotation $r : \Sigma_g \to \Sigma_g$ by $-\frac{2\pi}{g}$ about the $x$-axis and the simple closed curves $a_i,b_i,c_i$ on $\Sigma_g$ for $i=1,2,\ldots,g$.
• Figure 3: The simple closed curve $d_1$ and $d_2$ on $\Sigma_g$.
• Figure 4: The curves appearing in the proof of Theorem \ref{['thm:2']} and Proposition \ref{['prop:3']}.

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 2.1: Li
• Lemma 2.2: Ni
• proof
• Theorem 2.3: G, Grushko Theorem
• Lemma 2.4
• proof