Filling functions of arithmetic groups

Abstract

The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When $n$ is less than the rank of the associated symmetric space, we show that the $n$-dimensional filling volume function of the lattice grows at the same rate as that of the associated symmetric space, and when $n$ is equal to the rank, we show that the $n$-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky–Mozes–Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux–Wortman.

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      url = {https://doi.org/10.1007/PL00001641},
      zblnumber = {1037.53038},
      }
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    @ARTICLE{Leu-Kazh,
      author = {Leuzinger, Enrico},
      title = {Kazhdan's property ({T}), {$L^2$}-spectrum and isoperimetric inequalities for locally symmetric spaces},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {78},
      year = {2003},
      number = {1},
      pages = {116--133},
      issn = {0010-2571},
      mrclass = {53C24 (22E40 58J65)},
      mrnumber = {1966754},
      mrreviewer = {A. I. Danilenko},
      doi = {10.1007/s000140300005},
      url = {https://doi.org/10.1007/s000140300005},
      zblnumber = {1027.22015},
      }
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    @ARTICLE{LeuPoly,
      author = {Leuzinger, Enrico},
      title = {On polyhedral retracts and compactifications of locally symmetric spaces},
      journal = {Differential Geom. Appl.},
      fjournal = {Differential Geometry and its Applications},
      volume = {20},
      year = {2004},
      number = {3},
      pages = {293--318},
      issn = {0926-2245},
      mrclass = {53C35 (22E40 53C20)},
      mrnumber = {2053916},
      doi = {10.1016/j.difgeo.2004.03.001},
      url = {https://doi.org/10.1016/j.difgeo.2004.03.001},
      zblnumber = {1052.22008},
      }
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    @ARTICLE{LeuISO,
      author = {Leuzinger, Enrico},
      title = {Optimal higher-dimensional {D}ehn functions for some {$\rm CAT(0)$} lattices},
      journal = {Groups Geom. Dyn.},
      fjournal = {Groups, Geometry, and Dynamics},
      volume = {8},
      year = {2014},
      number = {2},
      pages = {441--466},
      issn = {1661-7207},
      mrclass = {20F67 (20E42 20F69 53C35)},
      mrnumber = {3231223},
      mrreviewer = {Mark F. Hagen},
      doi = {10.4171/GGD/233},
      url = {https://doi.org/10.4171/GGD/233},
      zblnumber = {1343.20046},
      }
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      author = {Lubotzky, Alexander and Mozes, Shahar and Raghunathan, M. S.},
      title = {The word and riemannian metrics on lattices of semisimple groups},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
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      author = {Leuzinger, Enrico and Pittet, C.},
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      journal = {Geom. Funct. Anal.},
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      mrnumber = {1392327},
      mrreviewer = {Raul Quiroga-Barranco},
      doi = {10.1007/BF02249261},
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    @ARTICLE{LeYoRank1,
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      journal = {Geom. Dedicata},
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      mrclass = {22E40 (20F65 53C35)},
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      year = {1973},
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      doi = {10.1090/S0002-9939-08-09203-4},
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      mrreviewer = {Enrico Leuzinger},
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    @ARTICLE{YoungHigherSol,
      author = {Young, Robert},
      title = {Lipschitz connectivity and filling invariants in solvable groups and buildings},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {18},
      year = {2014},
      number = {4},
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      issn = {1465-3060},
      mrclass = {20F65 (20E42 20F16)},
      mrnumber = {3268779},
      mrreviewer = {Matthew C. B. Zaremsky},
      doi = {10.2140/gt.2014.18.2375},
      url = {https://doi.org/10.2140/gt.2014.18.2375},
      zblnumber = {1347.20046},
      }

Authors

Enrico Leuzinger

Karlsruher Institut für Technologie, Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruhe, Germany

Robert Young

Courant Institute of Mathematical Sciences, New York, NY