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Transactions of the American Mathematical Society

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Integer hulls of linear polyhedra and scl in families


Authors: Danny Calegari and Alden Walker
Journal: Trans. Amer. Math. Soc. 365 (2013), 5085-5102
MSC (2010): Primary 11P21, 11H06, 57M07, 20F65, 20J05
DOI: https://doi.org/10.1090/S0002-9947-2013-05775-3
Published electronically: February 26, 2013
MathSciNet review: 3074368
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Abstract | References | Similar Articles | Additional Information

Abstract: The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size $ O(n)$ have eventually quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is eventually a ratio of quasipolynomials, and that unit balls in the scl norm eventually quasiconverge in finite-dimensional surgery families.


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  • 1. Bradford Barber, David Dobkin, and Hannu Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Software 22 (1996), no. 4, 469-483. MR 1428265 (97g:65292)
  • 2. Alexander Barvinok, Integer points in polyhedra, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2455889
  • 3. Matthias Beck, A closer look at lattice points in rational simplices, Electron. J. Combin. 6 (1999), Research Paper 37, 9 pp. (electronic). MR 1725701 (2001b:11025)
  • 4. Matthias Beck and Sinai Robins, Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, New York, 2007, Integer-point enumeration in polyhedra. MR 2271992 (2007h:11119)
  • 5. Danny Calegari, scl, MSJ Memoirs, vol. 20, Mathematical Society of Japan, Tokyo, 2009. MR 2527432
  • 6. -, scl, sails, and surgery, J. Topol. 4 (2011), no. 2, 305-326. MR 2805993
  • 7. Danny Calegari and Alden Walker, scabble, computer program available from the second author's website.
  • 8. Sheng Chen, Nan Li, and Steven Sam, Generalized Ehrhart polynomials, Trans. AMS (2012), no. 364, 551-569. MR 2833591
  • 9. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206 (94i:11105)
  • 10. D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of $ 3$-manifolds, Invent. Math. 118 (1994), no. 1, 47-84. MR 1288467 (95g:57029)
  • 11. Vladimir Fock and Alexander Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006), no. 103, 1-211. MR 2233852 (2009k:32011)
  • 12. Stavros Garoufalidis, The degree of a $ q$-holonomic sequence is a quadratic quasi-polynomial, Electron. J. Combin. 18 (2011), no. 2, Paper 4, 23. MR 2795781
  • 13. Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354 (2002k:55001)
  • 14. András Juhász, The sutured Floer homology polytope, Geom. Topol. 14 (2010), no. 3, 1303-1354. MR 2653728 (2011i:57042)
  • 15. Hendrik Lenstra, Integer programming with a fixed number of variables, Math. Oper. Res. 8 (1983), no. 4, 538-548. MR 727410 (86f:90106)
  • 16. Richard Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997, with a foreword by Gian-Carlo Rota. Corrected reprint of the 1986 original. MR 1442260 (98a:05001)
  • 17. Bernd Sturmfels and Rekha Thomas, Variation of cost functions in integer programming, Math. Programming 77 (1997), no. 3, Ser. A, 357-387. MR 1447563 (98e:90099)
  • 18. William P. Thurston, A norm for the homology of $ 3$-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339, i-vi and 99-130. MR 823443 (88h:57014)
  • 19. -, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417-431. MR 956596 (89k:57023)
  • 20. Nikolai Zolotykh, On the number of vertices in integer linear programming problems, arXiv:math/0611356v1 [math.CO].

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Additional Information

Danny Calegari
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Address at time of publication: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: dannyc@its.caltech.edu, dannyc@math.uchicago.edu

Alden Walker
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Address at time of publication: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: awalker@caltech.edu, akwalker@math.uchicago.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05775-3
Received by editor(s): November 22, 2010
Received by editor(s) in revised form: November 29, 2011
Published electronically: February 26, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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