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

Lvzhou Chen, Nicolaus Heuer

Published 2019-10-30Version 1

The 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 compute the exact gap of graph products. We prove these statements by characterizing maps of surfaces to a suitable K(G,1). In many cases, we also find families of quasimorphisms that realize these gaps. In particular, we construct quasimorphisms realizing the 1/2-gap for free groups explicitly induced by actions on the circle.