2019/10/30 by Lvzhou Chen, Chen, Lvzhou, Nicolaus Heuer +1 · 2 citations
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1910.14146
openalex publication_date 2019/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Stable commutator length sclG(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 sclG(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 sclG 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 sclG(g)>=C or sclG(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.