대한수학회지 (Journal of the Korean Mathematical Society)

  • 제57권5호
  • /
  • Pages.1103-1118
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  • 2020
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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GEOMETRIC SIMPLICIAL EMBEDDINGS OF ARC-TYPE GRAPHS

  • 투고 : 2019.06.12
  • 심사 : 2019.12.16
  • 발행 : 2020.09.01
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초록

In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These multiarc graphs naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certain complexity conditions, that simplicial maps between them only arise in the "obvious way". We also observe that, again under necessary complexity conditions, subsurface strata are convex. Put together, these results imply that certain simplicial maps always give rise to convex images.

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참고문헌

  1. J. Aramayona, Simplicial embeddings between pants graphs, Geom. Dedicata 144 (2010), 115-128. https://doi.org/10.1007/s10711-009-9391-0
  2. J. Aramayona, T. Koberda, and H. Parlier, Injective maps between flip graphs, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 5, 2037-2055. http://aif.cedram.org/item?id=AIF_2015__65_5_2037_0 https://doi.org/10.5802/aif.2981
  3. J. Aramayona, C. Lecuire, H. Parlier, and K. J. Shackleton, Convexity of strata in diagonal pants graphs of surfaces, Publ. Mat. 57 (2013), no. 1, 219-237. https://doi.org/10.5565/PUBLMAT_57113_08
  4. J. Aramayona, H. Parlier, and K. J. Shackleton, Totally geodesic subgraphs of the pants complex, Math. Res. Lett. 15 (2008), no. 2, 309-320. https://doi.org/10.4310/MRL.2008.v15.n2.a9
  5. J. Aramayona, H. Parlier, and K. J. Shackleton, Constructing convex planes in the pants complex, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3523-3531. https://doi.org/10.1090/S0002-9939-09-09907-9
  6. J. Aramayona and J. Souto, Homomorphisms between mapping class groups, Geom. Topol. 16 (2012), no. 4, 2285-2341. https://doi.org/10.2140/gt.2012.16.2285
  7. J. F. Brock, The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003), no. 3, 495-535. https://doi.org/10.1090/S0894-0347-03-00424-7
  8. V. Disarlo, Combinatorial rigidity of arc complexes, preprint.
  9. V. Disarlo and H. Parlier, The geometry of flip graphs and mapping class groups, Trans. Amer. Math. Soc. 372 (2019), no. 6, 3809-3844. https://doi.org/10.1090/tran/7356
  10. V. Erlandsson and F. Fanoni, Simplicial embeddings between multicurve graphs, Michigan Math. J. 66 (2017), no. 3, 549-567. https://doi.org/10.1307/mmj/1496995339
  11. E. Irmak and J. D. McCarthy, Injective simplicial maps of the arc complex, Turkish J. Math. 34 (2010), no. 3, 339-354.
  12. M. Korkmaz and A. Papadopoulos, On the ideal triangulation graph of a punctured surface, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1367-1382. https://doi.org/10.5802/aif.2725
  13. S. J. Taylor and A. Zupan, Products of Farey graphs are totally geodesic in the pants graph, J. Topol. Anal. 8 (2016), no. 2, 287-311. https://doi.org/10.1142/S1793525316500096
  14. S. A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275-296. http://projecteuclid.org/euclid.jdg/1214440853 https://doi.org/10.4310/jdg/1214440853