Spectral gap of scl in graphs of groups and -manifolds
arXiv:1910.14146
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.
arXiv commentv3: Significantly shortened for publication purposes. Improved the exposition throughout and addressed comments from referees. Main results about spectral gaps in graphs of groups, 3-manifolds, and graph products remain unchanged. Deleted parts will appear in separate papers. Replaced the incorrect Lemma 8.6 in v2 by Lemma 6.6 in v3 without affecting other results. 50 pages and 8 figures