[Paper Review] Spectral gap of scl in graphs of groups and $3$-manifolds

Lvzhou Chen, Nicolaus Heuer|arXiv (Cornell University)|Oct 30, 2019

Geometric and Algebraic Topology32 references4 citations

TL;DR

This paper establishes sharp spectral gaps for stable commutator length (scl) in groups acting on trees, particularly graphs of groups and 3-manifold groups. It proves that for hyperbolic elements in such groups, scl is bounded below by $1/2 - 1/n$ when edge stabilizers are $n$-relatively torsion-free in vertex stabilizers, with a sharp $1/2$ gap when $n = \infty$. The results are derived via surface maps to $K(G,1)$ complexes and quasimorphism constructions, and are applied to show spectral gaps in fundamental groups of closed 3-manifolds via JSJ decompositions.

ABSTRACT

Stable commutator length scl_G(g) of an element g in a group G is an invariant for group elements sensitive to the geometry and dynamics of G. For any group G acting on a tree, we prove a sharp bound scl_G(g)>=1/2 for any g acting without fixed points, provided that the stabilizer of each edge is relatively torsion-free in its vertex stabilizers. The sharp gap becomes 1/2-1/n if the edge stabilizers are n-relatively torsion-free in vertex stabilizers. We also compute scl_G for elements acting with a fixed point. This implies many such groups have a spectral gap, that is, there is a constant C>0 such that either scl_G(g)>=C or scl_G(g)=0. New examples include the fundamental group of any 3-manifold using the JSJ decomposition, though the gap must depend on the manifold. We also obtain the optimal spectral gap of graph products of group without 2-torsion. We prove these statements by characterizing maps of surfaces to a suitable K(G,1). For groups acting on trees, we also construct explicit quasimorphisms and apply Bavard's duality to give a different proof of our spectral gap theorem under stronger assumptions.

Motivation & Objective

• To establish sharp lower bounds on stable commutator length (scl) for hyperbolic elements in groups acting on trees with controlled edge stabilizers.
• To characterize scl for elliptic elements in terms of vertex group scl and edge stabilizer adjustments.
• To prove the existence of spectral gaps in fundamental groups of closed 3-manifolds using JSJ decompositions and geometrization.
• To construct explicit homogeneous quasimorphisms realizing the $1/2$-gap in free groups via circle actions.
• To generalize and strengthen prior spectral gap results for graphs of groups, including those in [CFL16], [Che18b], [DH91], and [Heu19].

Proposed method

• Use of surface maps to $K(G,1)$ complexes to characterize scl, particularly by lifting surface maps to finite covers and analyzing their topology.
• Application of pleated surface theory and hyperbolic geometry, including the $2$-dimensional Margulis lemma and thin-thick decomposition of surfaces.
• Construction of explicit quasimorphisms on free groups via circle actions, with defect $\leq 1$ and value $\geq 1$ on hyperbolic elements.
• Reduction of scl computation for elliptic chains to infima over adjusted chains in vertex groups, accounting for edge stabilizer contributions.
• Use of Dehn twists and compression to ensure surface maps do not factor through circles, preserving geometric control.
• Analysis of conjugate pairs of hyperbolic elements and their commutators to exclude small translation lengths, leveraging uniform lower bounds in von Dyck groups.

Experimental results

Research questions

• RQ1 What is the sharp lower bound for stable commutator length of hyperbolic elements in a group acting on a tree with $n$-relatively torsion-free edge stabilizers?
• RQ2 How does the spectral gap in scl depend on the relative torsion-freeness condition of edge stabilizers in vertex stabilizers?
• RQ3 Can the spectral gap in fundamental groups of closed 3-manifolds be established using JSJ decompositions and geometric methods?
• RQ4 What is the exact value of $\mathrm{scl}_G(t)$ for an element $t$ in an amalgamated free product $G = A \star_\mathbb{Z} B$?
• RQ5 Can explicit quasimorphisms be constructed that realize the $1/2$-gap in free groups?

Key findings

• For any hyperbolic element $g$ in a group $G$ acting on a tree, $\mathrm{scl}_G(g) \geq 1/2 - 1/n$ if edge stabilizers are $n$-relatively torsion-free in vertex stabilizers, with equality possible.
• When edge stabilizers are relatively torsion-free ($n = \infty$), the sharp lower bound is $\mathrm{scl}_G(g) \geq 1/2$, which is optimal.
• For any closed oriented connected 3-manifold $M$, $\pi_1(M)$ has a spectral gap $C(M) > 0$, with $C(M)$ depending on $M$; this is proven via JSJ decomposition and geometrization.
• The scl of a chain of elliptic elements in a graph of groups is given by the infimum of the sum of scl values in vertex groups after adjusting by chains in edge stabilizers.
• For $G = A \star_\mathbb{Z} B$, $\mathrm{scl}_G(t) = \min(\mathrm{scl}_A(t), \mathrm{scl}_B(t))$ when $t$ generates the amalgamated $\mathbb{Z}$.
• Explicit homogeneous quasimorphisms on free groups realize the $1/2$-gap, with defect $\leq 1$ and value $\geq 1$ on hyperbolic elements, constructed from circle actions.