If S is a genus g surface with b boundary components, so that 3g − 3 + b 2, then the curve complex C(S) has a vertex for each isotopy class of essential non-peripheral simple closed curves in S and a k-simplex for each collection of k + 1 disjoint vertices having disjoint representatives. By regarding each simplex as a Euclidean simplex of side-length one, C(S) turns out to be Gromov hyperbolic ([Invent. Math. 138, No.1, 103-149 (1999; Zbl 0941.32012)]). The present paper proves that, if the surface S has exactly one boundary component and genus two or more, than for each vertex ! 2 C(S) and for any r 2 N, the subcomplex spanned by C0(S) − B(!, r) is connected (where B(!, r) denotes the ball of radius r about the vertex !). In order to prove the above result, the author makes use of the fact that the complex of curves has no dead ends (Prop. 3.1 of this paper) and of the so called Birman short exact sequence (see [Annals of Mathematics Studies 82, Princeton (1975; Zbl 0305.57013)] and [Acta Math. 146, 231-270 (1981; Zbl 0477.32024)]). Note that, for the considered surfaces, the above result directly answers a question of Masur’s, and answers a question of G.Bell and K.Fujiara ([J. Lond. Math. Soc., II. Ser. 77, No. 1, 33-50 (2008; Zbl 1135.57010)]) in the negative. It is also evidence for a positive answer to a question of P.Storm (already verified in an independent way by Gabai in [Geom. Topol. 13, No. 2, 1017-1041 (2009; Zbl 1165.57015)]).
|Data di pubblicazione:||2012|
|Titolo:||REVIEW OF: "Schleimer Saul, The end of the curve complex, Groups Geom. Dyn. 5, No. 1, 169-176 (2011)". [DE059733321]|
|Autori:||Casali, Maria Rita|
|Autori del volume:||Schleimer, Saul|
|Appare nelle tipologie:||Recensione in Rivista|
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