Schleimer's hyperbolicity conjecture for multiarc and curve graphs

Let Σ\Sigma be a compact, connected, orientable surface, and let A\mathcal A be a multiarc and curve graph on Σ\Sigma. A compact, essential subsurface WΣW\subset\Sigma is a witness for A\mathcal A if W≇Σ03W\not\cong\Sigma_0^3 and every arc and curve system in V(A)V(\mathcal A) intersects WW. Let i(a,b)i(a,b) denote geometric intersection number, and let M0M\geq 0. Assume that (a,b)E(A)(a,b)\in E(\mathcal A) if and only if i(a,b)Mi(a,b)\leq M. Schleimer's hyperbolicity conjecture. The graph A\mathcal A is δ\delta-hyperbolic if and only if it does not admit disjoint connected witnesses. This conjecture proposes a geometric characterization of hyperbolicity for multiarc and curve graphs; its resolution status is not specified in the source.

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Primary source

Michael C. Kopreski, “Multiarc and curve graphs are hierarchically hyperbolic”, arXiv:2311.04356 (2024).

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