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Spectral gap of scl in graphs of groups and $3$-manifolds

Lvzhou Chen, Nicolaus Heuer

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.

Spectral gap of scl in graphs of groups and $3$-manifolds

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.

Paper Structure

This paper contains 29 sections, 70 theorems, 102 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Method
  3. Stable commutator length via quasimorphisms
  4. Stable commutator length via admissible surfaces
  5. Graph products
  6. $3$-manifold groups
  7. Background
  8. Stable commutator length
  9. Quasimorphisms and Bavard's duality
  10. Relative stable commutator length
  11. Bounded cohomology and circle actions
  12. Graphs of groups
  13. Scl of hyperbolic elements
  14. Surfaces in graphs of groups
  15. Lower bounds from linear programming duality
  16. ...and 14 more sections

Key Result

Theorem 1

Let $G$ be a group acting on a tree such that the stabilizer of every edge is $n$-RTF in the stabilizers of the its vertices. If $g \in G$ is hyperbolic, then If the stabilizer of every edge lies left relatively convex in the stabilizers of its vertices, then there is an explicit homogeneous quasimorphism $\phi$ (see Subsection subsubsec:scl via quasimorphisms for the definition) on $G$ such that

Figures (7)

  • Figure 2: On the bottom left we depict the thickened vertex space $N(X_v)$ for the unique vertex space $X_v$ in the realization $X$ (upper left) of $\mathrm{BS}(m,\ell)$; on the right depict a thickened vertex space in a more general situation, where the red circles are the edge spaces that we cut along.
  • Figure 3: A loop $\gamma$ trivially backtracks at an arc $a$ supported in the thickened vertex space $N(X_v)$ as shown on the left. It can be pushed off the vertex space $X_v$ by a homotopy as shown on the right.
  • Figure 4: Here are two possible components $\Sigma_1$ and $\Sigma_2$ of $S$ supported in some $N(X_v)$, where the blue parts are supported in $\underline{\gamma}$ and the red is part of $F$. The component $\Sigma_1$ has three boundary components: $\beta_1$ winds around an elliptic loop in $\underline{\gamma}$, the loop $\beta_2$ lies in $F$, and $\beta_3$ is a polygonal boundary. The component $\Sigma_2$ is a disk with polygonal boundary $\beta_4$, on which we have arcs $a_{i_1},a_{i_2},a_{i_3}$ in cyclic order.
  • Figure 5: Two paired turns on an edge space $X_e$, with arcs $a_1,a'_2\subset\gamma_i$ and $a'_1,a_2\subset\gamma_j$
  • Figure 6: Visualization of the summation in the case $L=6$, where the first equality uses the gluing condition $t_{ij}=t_{j-1,i+1}$
  • ...and 2 more figures

Theorems & Definitions (156)

  • Theorem 1: Theorems \ref{['thm: $n$-RTF gap, weak version']} and \ref{['thm:left relatively convex graph of groups']}
  • Theorem 2: Theorem \ref{['thm: vert scl compute']}
  • Theorem 3: Theorem \ref{['thm:3mfdgap']}
  • Theorem 4: Theorem \ref{['thm: gap for graph products, element-wise statement']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6: Clay--Forester--Louwsma
  • ...and 146 more