[Paper Review] Stable commutator length on mapping class groups

Mladen Bestvina, Kenneth Bromberg+1 more|arXiv (Cornell University)|Jun 11, 2013

Geometric and Algebraic Topology20 references6 citations

TL;DR

This paper establishes a precise criterion for when elements in finite-index subgroups of mapping class groups have positive stable commutator length (scl), using actions on quasi-trees derived from the BBF construction. It proves that scl(g) > 0 if and only if some chiral equivalence class of pure components in the Nielsen-Thurston decomposition is essential, and further shows that scl is uniformly bounded away from zero when positive, even in the Torelli group where scl(g) > 0 for all nontrivial g.

ABSTRACT

Let $Γ$ be a finite index subgroup of the mapping class group $MCG(Σ)$ of a closed orientable surface $Σ$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $g\inΓ$ has positive stable commutator length. In addition, we show that in these situations the stable commutator length, if nonzero, is uniformly bounded away from 0. The method works for certain subgroups of infinite index as well and we show $scl$ is uniformly positive on the nontrivial elements of the Torelli group. The proofs use our earlier construction in the paper "Constructing group actions on quasi-trees and applications to mapping class groups" of group actions on quasi-trees.

Motivation & Objective

• To determine precisely when elements in finite-index subgroups of mapping class groups have positive stable commutator length (scl).
• To extend the understanding of scl beyond hyperbolic groups by identifying new obstructions and conditions in mapping class groups.
• To prove that scl is uniformly bounded away from zero for non-vanishing scl in finite-index subgroups, including the Torelli group.
• To establish that every nontrivial element of the Torelli group has positive scl, resolving a key case in the theory.

Proposed method

• The authors use the BBF construction of group actions on quasi-trees to analyze the dynamics of mapping class group elements.
• They decompose elements via the Nielsen-Thurston decomposition into pseudo-Anosov and Dehn twist components, focusing on chiral equivalence classes.
• A chiral equivalence class is deemed essential if, after conjugation to a common supporting subsurface, the product of its powers has infinite order.
• The proof relies on quasi-morphisms constructed from the quasi-tree actions, linking dynamical behavior to scl positivity.
• The method applies to both finite-index subgroups and infinite-index subgroups such as the Torelli group.
• The authors use homological and topological constraints (e.g., homology classes over Z_p) to rule out certain conjugacy and orbit behaviors.

Experimental results

Research questions

• RQ1 When does an element g in a finite-index subgroup G of MCG(Σ) have scl(g) > 0?
• RQ2 What dynamical or algebraic conditions on g ensure scl(g) > 0, beyond finite order or achirality?
• RQ3 Is scl uniformly bounded away from zero for non-vanishing scl in finite-index subgroups of MCG(Σ)?
• RQ4 Does every nontrivial element of the Torelli group have positive stable commutator length?
• RQ5 Can the BBF quasi-tree construction be used to detect scl positivity in subgroups of MCG(Σ)?

Key findings

• scl(g) > 0 if and only if some chiral equivalence class of pure components in the Nielsen-Thurston decomposition of g is essential.
• For any finite-index subgroup G < MCG(Σ), there exists ε = ε(G) > 0 such that scl_G(g) > ε whenever scl_G(g) > 0.
• In the Torelli group T, scl_T(g) > 0 for every nontrivial element g ∈ T.
• For any finite-index subgroup G < MCG(Σ), if g has exponential growth, then scl_G(g) > 0.
• The sequence cl_G(g^n) is bounded if and only if scl_G(g) = 0.
• Every nontrivial multitwist in the Torelli group has positive scl, even when supported on nonseparating curves, due to nontrivial abelianization.