Spectral gap of scl in graphs of groups and $3$-manifolds
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arXiv
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| Format: | Preprint |
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2019
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| author | Chen, Lvzhou Heuer, Nicolaus |
| author_facet | Chen, Lvzhou Heuer, Nicolaus |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_14146 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Spectral gap of scl in graphs of groups and $3$-manifolds Chen, Lvzhou Heuer, Nicolaus Geometric Topology Group Theory 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. |
| title | Spectral gap of scl in graphs of groups and $3$-manifolds |
| topic | Geometric Topology Group Theory |
| url | https://arxiv.org/abs/1910.14146 |