Stable commutator lengths of integral chains in right-angled Artin groups

Abstract: It follows from theorems of Agol and Kahn-Markovic that the fundamental group of any closed hyperbolic 3-manifold contains a special subgroup of finite index. Very little is known about how large the index needs to be. Motivated by this, in this joint work with Nicolaus Heuer, we study stable commutator lengths (scl) of integral chains in right-angled Artin groups (RAAGs). Topologically, an integral 1-chain in a group G is a collection of loops in the K(G,1) space with integral weights, and its scl is the least complexity of surfaces bounding the weighted loops. We show that the infimal positive scl of integral chains in any RAAG is positive, and its size explicitly depends on the defining graph of the RAAG up to a multiplicative constant 12. In particular, the size is non-uniform among RAAGs, which is unexpected.

algebraic topologydifferential geometrygeneral topologygroup theorygeometric topology

Audience: researchers in the topic

Ohio State Topology and Geometric Group Theory Seminar

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