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A collection of nonfilling multicurve complexes. (English) Zbl 1325.57011

For any connected, orientable compact surface \(S=S_{g,n}\), with \(3g - 3 +n \geq 2\), and for any integer \(k\), with \(1 \leq k \leq 3g - 5 +n\), the author introduces the notion of \(k\)–Curve Complex of \(S\), denoted by \(k-C(S)\): \(k-C(S)\) is a simplicial complex whose vertices are the multicurves of cardinality \(k\) (i.e. \(k\)-tuples of isotopy classes of disjoint non-peripheral, simple closed curves) in \(S\), and where two vertices are joined by an edge if the corresponding multicurves do not fill \(S\) and are not isotopic. An \(r\)-simplex (\(r \geq2 \)) in \(k-C(S)\) is realized when a collection of \(r+1\) multicurves, of cardinality \(k\), satisfy the two conditions pairwise.
Roughly speaking, the \(k\)-Curve Complex is a new complex, which extends the classical notion of curve complex by generalizing the notion of disjointness with the notion of non-fillingness, in order to interpolate the 1-skeleton of the curve complex and the pants graph of a given surface.

The author proves that each \(k\)-Curve Complex is connected, by making use of the hierarchy machinery developed by H. A. Masur and Y. N. Minsky [Geom. Funct. Anal. 10, No. 4, 902–974 (2000; Zbl 0972.32011)]. Moreover, the coarse geometry of this collection of complexes is studied: in particular, \(1-C(S)\) is proved to be hyperbolic, while distances in \(k-C(S)\) are related to the distances in the marking complex of \(S\), and hence to the length of word path in the mapping class group \(\mathcal{MCG}(S)\).
Finally, the paper provides an improved bound for distance of Heegaard splittings of a 3-manifold, by using the distance in the 1-curve complex, \(1-C(S)\), \(S\) being the (closed) splitting surface.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57Q05 General topology of complexes
57M99 General low-dimensional topology

Citations:

Zbl 0972.32011
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References:

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