Abstract
We develop a new criterion to tell if a group G has the maximal gap of 1/2 in stable commutator length (scl). For amalgamated free products \({G = A \star_C B}\) we show that every element g in the commutator subgroup of G which does not conjugate into A or B satisfies \({{\rm scl}(g) \geq 1/2}\), provided that C embeds as a left relatively convex subgroup in both A and B. We deduce from this that every non-trivial element g in the commutator subgroup of a right-angled Artin group G satisfies \({{\rm scl}(g) \geq 1/2}\). This bound is sharp and is inherited by all fundamental groups of special cube complexes. We prove these statements by constructing explicit extremal homogeneous quasimorphisms \({\bar{\phi} : G \to \mathbb{R}}\) satisfying \({\bar{\phi}(g) \geq 1}\) and \({D(\bar{\phi})\leq 1}\). Such maps were previously unknown, even for non-abelian free groups. For these quasimorphisms \({\bar{\phi}}\) there is an action \({\rho : G \to \mathrm{Homeo}^+(S^1)}\) on the circle such that \({[\delta^1 \bar{\phi}]=\rho^*{\rm eu}^\mathbb{R}_b \in {\rm H}^2_b(G,\mathbb{R})}\), for \({{\rm eu}^\mathbb{R}_b}\) the real bounded Euler class.
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Acknowledgments
I would like to thank my supervisor, Martin Bridson, for his help, support and guidance, and Ric Wade for his very helpful comments. I would further like to thank the referee for carefully reading the paper and recommending helpful improvements. Moreover, I would like to thank the Isaac Newton Institute for Mathematical Sciences in Cambridge for support and hospitality during the programme Non-Positive Curvature Group Actions and Cohomology where work on this paper was undertaken. I would like to thank Danny Calegari for a stimulating conversation at the Isaac Newton Institute and Max Forester for pointing out errors in a previous version of this paper. This work was supported by EPSRC grant no EP/K032208/1. The author is also supported by the Oxford-Cocker Scholarship.
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Heuer, N. Gaps in scl for Amalgamated Free Products and RAAGs. Geom. Funct. Anal. 29, 198–237 (2019). https://doi.org/10.1007/s00039-019-00477-5
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DOI: https://doi.org/10.1007/s00039-019-00477-5