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.
-
[AbramBrown]
P. Abramenko and K. S. Brown, Buildings, Springer, New York, 2008, vol. 248.@BOOK{AbramBrown,
author = {Abramenko, Peter and Brown, Kenneth S.},
title = {Buildings},
series = {Grad. Texts in Math.},
volume = {248},
note = {Theory and applications},
publisher = {Springer, New York},
year = {2008},
pages = {xxii+747},
isbn = {978-0-387-78834-0},
mrclass = {20E42 (20F55 20J06 51E24 51F15)},
mrnumber = {2439729},
mrreviewer = {Ralf Koehl},
doi = {10.1007/978-0-387-78835-7},
url = {https://doi.org/10.1007/978-0-387-78835-7},
zblnumber = {1214.20033},
} -
[ABDDYraag] A. Abrams, N. Brady, P. Dani, M. Duchin, and R. Young, "Pushing fillings in right-angled Artin groups," J. Lond. Math. Soc. (2), vol. 87, iss. 3, pp. 663-688, 2013.
@ARTICLE{ABDDYraag,
author = {Abrams, Aaron and Brady, Noel and Dani, Pallavi and Duchin, Moon and Young, Robert},
title = {Pushing fillings in right-angled {A}rtin groups},
journal = {J. Lond. Math. Soc. (2)},
fjournal = {Journal of the London Mathematical Society. Second Series},
volume = {87},
year = {2013},
number = {3},
pages = {663--688},
issn = {0024-6107},
mrclass = {20F65 (28A75)},
mrnumber = {3073670},
mrreviewer = {Yago Antol\'ın},
doi = {10.1112/jlms/jds064},
url = {https://doi.org/10.1112/jlms/jds064},
zblnumber = {1294.20052},
} -
[ABDYpnas] A. Abrams, N. Brady, P. Dani, and R. Young, "Homological and homotopical Dehn functions are different," Proc. Natl. Acad. Sci. USA, vol. 110, iss. 48, pp. 19206-19212, 2013.
@ARTICLE{ABDYpnas,
author = {Abrams, Aaron and Brady, Noel and Dani, Pallavi and Young, Robert},
title = {Homological and homotopical {D}ehn functions are different},
journal = {Proc. Natl. Acad. Sci. USA},
fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
volume = {110},
year = {2013},
number = {48},
pages = {19206--19212},
issn = {0027-8424},
mrclass = {20F65 (57M07)},
mrnumber = {3153947},
mrreviewer = {Gilbert Levitt},
doi = {10.1073/pnas.1207377110},
url = {https://doi.org/10.1073/pnas.1207377110},
zblnumber = {1300.57003},
} -
[athreyaULTRA] J. S. Athreya, A. Ghosh, and A. Prasad, "Ultrametric logarithm laws, II," Monatsh. Math., vol. 167, iss. 3-4, pp. 333-356, 2012.
@ARTICLE{athreyaULTRA,
author = {Athreya, J. S. and Ghosh, Anish and Prasad, Amritanshu},
title = {Ultrametric logarithm laws, {II}},
journal = {Monatsh. Math.},
fjournal = {Monatshefte für Mathematik},
volume = {167},
year = {2012},
number = {3-4},
pages = {333--356},
issn = {0026-9255},
mrclass = {37D40 (11J83 11K60 37A17)},
mrnumber = {2961287},
mrreviewer = {Radhakrishnan Nair},
doi = {10.1007/s00605-012-0376-y},
url = {https://doi.org/10.1007/s00605-012-0376-y},
zblnumber = {1338.37007},
} -
[AWP] J. M. Alonso, X. Wang, and S. J. Pride, "Higher-dimensional isoperimetric (or Dehn) functions of groups," J. Group Theory, vol. 2, iss. 1, pp. 81-112, 1999.
@ARTICLE{AWP,
author = {Alonso, J. M. and Wang, X. and Pride, S. J.},
title = {Higher-dimensional isoperimetric (or {D}ehn) functions of groups},
journal = {J. Group Theory},
fjournal = {Journal of Group Theory},
volume = {2},
year = {1999},
number = {1},
pages = {81--112},
issn = {1433-5883},
mrclass = {20M20 (20F38 57M07)},
mrnumber = {1670329},
mrreviewer = {Wim Malfait},
doi = {10.1515/jgth.1999.008},
url = {https://doi.org/10.1515/jgth.1999.008},
zblnumber = {0927.20021},
} -
[BBFSsnowflake] N. Brady, M. R. Bridson, M. Forester, and K. Shankar, "Snowflake groups, Perron-Frobenius eigenvalues and isoperimetric spectra," Geom. Topol., vol. 13, iss. 1, pp. 141-187, 2009.
@ARTICLE{BBFSsnowflake,
author = {Brady, Noel and Bridson, Martin R. and Forester, Max and Shankar, Krishnan},
title = {Snowflake groups, {P}erron-{F}robenius eigenvalues and isoperimetric spectra},
journal = {Geom. Topol.},
fjournal = {Geometry \& Topology},
volume = {13},
year = {2009},
number = {1},
pages = {141--187},
issn = {1465-3060},
mrclass = {20F65 (20E06 20F69 57M07)},
mrnumber = {2469516},
mrreviewer = {Michel Coornaert},
doi = {10.2140/gt.2009.13.141},
url = {https://doi.org/10.2140/gt.2009.13.141},
zblnumber = {1228.20031},
} -
[BoissonnatDyerGhosh] J. Boissonnat, R. Dyer, and A. Ghosh, "Delaunay triangulation of manifolds," Found. Comput. Math., vol. 18, iss. 2, pp. 399-431, 2018.
@ARTICLE{BoissonnatDyerGhosh,
author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit},
title = {Delaunay triangulation of manifolds},
journal = {Found. Comput. Math.},
fjournal = {Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics},
volume = {18},
year = {2018},
number = {2},
pages = {399--431},
issn = {1615-3375},
mrclass = {57R05 (52B70 54B15)},
mrnumber = {3777784},
mrreviewer = {Vladimir Aleksandrovich Klyachin},
doi = {10.1007/s10208-017-9344-1},
url = {https://doi.org/10.1007/s10208-017-9344-1},
zblnumber = {1395.57032},
} -
[BestvinaEskinWortman] M. Bestvina, A. Eskin, and K. Wortman, "Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups," J. Eur. Math. Soc. (JEMS), vol. 15, iss. 6, pp. 2165-2195, 2013.
@ARTICLE{BestvinaEskinWortman,
author = {Bestvina, Mladen and Eskin, Alex and Wortman, Kevin},
title = {Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups},
journal = {J. Eur. Math. Soc. (JEMS)},
fjournal = {Journal of the European Mathematical Society (JEMS)},
volume = {15},
year = {2013},
number = {6},
pages = {2165--2195},
issn = {1435-9855},
mrclass = {11F75 (11F06)},
mrnumber = {3120741},
mrreviewer = {Enrico Leuzinger},
doi = {10.4171/JEMS/419},
url = {https://doi.org/10.4171/JEMS/419},
zblnumber = {1372.11055},
} -
[BuxKohlWitzel] K. Bux, R. Köhl, and S. Witzel, "Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem," Ann. of Math. (2), vol. 177, iss. 1, pp. 311-366, 2013.
@ARTICLE{BuxKohlWitzel,
author = {Bux, Kai-Uwe and Köhl, Ralf and Witzel, Stefan},
title = {Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {177},
year = {2013},
number = {1},
pages = {311--366},
issn = {0003-486X},
mrclass = {20G30 (20E42 22E40)},
mrnumber = {2999042},
mrreviewer = {Alain Valette},
doi = {10.4007/annals.2013.177.1.6},
url = {https://doi.org/10.4007/annals.2013.177.1.6},
zblnumber = {1290.20039},
} -
[BridsonWordProblem] M. R. Bridson, "The geometry of the word problem," in Invitations to Geometry and Topology, Oxford Univ. Press, Oxford, 2002, vol. 7, pp. 29-91.
@INCOLLECTION{BridsonWordProblem,
author = {Bridson, Martin R.},
title = {The geometry of the word problem},
booktitle = {Invitations to Geometry and Topology},
series = {Oxford Grad. Texts Math.},
volume = {7},
pages = {29--91},
publisher = {Oxford Univ. Press, Oxford},
year = {2002},
mrclass = {20F65 (20F10 49J05 57M07)},
mrnumber = {1967746},
mrreviewer = {Ross Geoghegan},
zblnumber = {0996.54507},
} -
[BuxWortFiniteness] K. Bux and K. Wortman, "Finiteness properties of arithmetic groups over function fields," Invent. Math., vol. 167, iss. 2, pp. 355-378, 2007.
@ARTICLE{BuxWortFiniteness,
author = {Bux, Kai-Uwe and Wortman, Kevin},
title = {Finiteness properties of arithmetic groups over function fields},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {167},
year = {2007},
number = {2},
pages = {355--378},
issn = {0020-9910},
mrclass = {11F75 (11F06 20G35 22E40 57M07)},
mrnumber = {2270455},
mrreviewer = {Herbert Abels},
doi = {10.1007/s00222-006-0017-y},
url = {https://doi.org/10.1007/s00222-006-0017-y},
zblnumber = {1126.20030},
} -
[CohenSP] D. B. Cohen, "A Dehn function for ${ Sp}(2n,\Bbb Z)$," J. Topol. Anal., vol. 9, iss. 2, pp. 225-290, 2017.
@ARTICLE{CohenSP,
author = {Cohen, David Bruce},
title = {A {D}ehn function for {${\rm Sp}(2n,\Bbb Z)$}},
journal = {J. Topol. Anal.},
fjournal = {Journal of Topology and Analysis},
volume = {9},
year = {2017},
number = {2},
pages = {225--290},
issn = {1793-5253},
mrclass = {20F65 (20G30 22E40 57M07)},
mrnumber = {3622234},
mrreviewer = {Bruno P. Zimmermann},
doi = {10.1142/S1793525317500121},
url = {https://doi.org/10.1142/S1793525317500121},
zblnumber = {1400.20035},
} -
[CohenLipOne] D. B. Cohen, "Lipschitz 1-connectedness for some solvable Lie groups," Canad. J. Math., vol. 71, iss. 3, pp. 533-555, 2019.
@ARTICLE{CohenLipOne,
author = {Cohen, David Bruce},
title = {Lipschitz 1-connectedness for some solvable {L}ie groups},
journal = {Canad. J. Math.},
fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
volume = {71},
year = {2019},
number = {3},
pages = {533--555},
issn = {0008-414X},
mrclass = {20F65 (22E25)},
mrnumber = {3952489},
doi = {10.4153/cjm-2017-038-8},
url = {https://doi.org/10.4153/cjm-2017-038-8},
zblnumber = {07061565},
} -
[CorTessSolv] Y. Cornulier and R. Tessera, "Geometric presentations of Lie groups and their Dehn functions," Publ. Math. Inst. Hautes Études Sci., vol. 125, pp. 79-219, 2017.
@ARTICLE{CorTessSolv,
author = {Cornulier, Yves and Tessera, Romain},
title = {Geometric presentations of {L}ie groups and their {D}ehn functions},
journal = {Publ. Math. Inst. Hautes \'{E}tudes Sci.},
fjournal = {Publications Mathématiques. Institut de Hautes \'{E}tudes Scientifiques},
volume = {125},
year = {2017},
pages = {79--219},
issn = {0073-8301},
mrclass = {17B70 (22E35)},
mrnumber = {3668649},
doi = {10.1007/s10240-016-0087-3},
url = {https://doi.org/10.1007/s10240-016-0087-3},
zblnumber = {1428.22012},
} -
[DrutuFilling] C. Dructu, "Filling in solvable groups and in lattices in semisimple groups," Topology, vol. 43, iss. 5, pp. 983-1033, 2004.
@ARTICLE{DrutuFilling,
author = {Dru\c{t}u, Cornelia},
title = {Filling in solvable groups and in lattices in semisimple groups},
journal = {Topology},
fjournal = {Topology. An International Journal of Mathematics},
volume = {43},
year = {2004},
number = {5},
pages = {983--1033},
issn = {0040-9383},
mrclass = {20F10 (20F65 22E25 22E40 51E24)},
mrnumber = {2079992},
mrreviewer = {Vitaly A. Roman\cprime kov},
doi = {10.1016/j.top.2003.11.004},
url = {https://doi.org/10.1016/j.top.2003.11.004},
zblnumber = {1083.20033},
} -
[ECHLPT] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word Processing in Groups, Jones and Bartlett Publishers, Boston, MA, 1992.
@BOOK{ECHLPT,
author = {Epstein, David B. A. and Cannon, James W. and Holt, Derek F. and Levy, Silvio V. F. and Paterson, Michael S. and Thurston, William P.},
title = {Word Processing in Groups},
publisher = {Jones and Bartlett Publishers, Boston, MA},
year = {1992},
pages = {xii+330},
isbn = {0-86720-244-0},
mrclass = {20F10 (03D40 20-02 68Q70)},
mrnumber = {1161694},
mrreviewer = {Richard M. Thomas},
zblnumber = {0764.20017},
} -
[FedFlemNormInt] H. Federer and W. H. Fleming, "Normal and integral currents," Ann. of Math. (2), vol. 72, pp. 458-520, 1960.
@ARTICLE{FedFlemNormInt,
author = {Federer, Herbert and Fleming, Wendell H.},
title = {Normal and integral currents},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {72},
year = {1960},
pages = {458--520},
issn = {0003-486X},
mrclass = {28.80 (53.45)},
mrnumber = {0123260},
mrreviewer = {L. C. Young},
doi = {10.2307/1970227},
url = {https://doi.org/10.2307/1970227},
zblnumber = {0187.31301},
} -
[GerstenSurv] S. M. Gersten, "Isoperimetric and isodiametric functions of finite presentations," in Geometric Group Theory, Vol. 1, Cambridge Univ. Press, Cambridge, 1993, vol. 181, pp. 79-96.
@INCOLLECTION{GerstenSurv,
author = {Gersten, Steve M.},
title = {Isoperimetric and isodiametric functions of finite presentations},
booktitle = {Geometric Group Theory, {V}ol. 1},
venue = {{S}ussex, 1991},
series = {London Math. Soc. Lecture Note Ser.},
volume = {181},
pages = {79--96},
publisher = {Cambridge Univ. Press, Cambridge},
year = {1993},
mrclass = {20F06 (57M07)},
mrnumber = {1238517},
mrreviewer = {John Meier},
doi = {10.1017/CBO9780511661860.008},
url = {https://doi.org/10.1017/CBO9780511661860.008},
zblnumber = {0829.20054},
} -
[GroFRM] M. Gromov, "Filling Riemannian manifolds," J. Differential Geom., vol. 18, iss. 1, pp. 1-147, 1983.
@ARTICLE{GroFRM,
author = {Gromov, Mikhael},
title = {Filling {R}iemannian manifolds},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {18},
year = {1983},
number = {1},
pages = {1--147},
issn = {0022-040X},
mrclass = {53C20 (53C21 57R99)},
mrnumber = {0697984},
mrreviewer = {Yu. Burago},
zblnumber = {0515.53037},
doi = {10.4310/jdg/1214509283},
url = {https://doi.org/10.4310/jdg/1214509283},
} -
[GroAII] M. Gromov, "Asymptotic invariants of infinite groups," in Geometric Group Theory, Vol. 2, Cambridge Univ. Press, Cambridge, 1993, vol. 182, pp. 1-295.
@INCOLLECTION{GroAII,
author = {Gromov, Mikhael},
title = {Asymptotic invariants of infinite groups},
booktitle = {Geometric Group Theory, {V}ol. 2},
venue = {{S}ussex, {U}university, July 14--19, 1991},
series = {London Math. Soc. Lecture Note Ser.},
volume = {182},
pages = {1--295},
publisher = {Cambridge Univ. Press, Cambridge},
year = {1993},
mrclass = {20F32 (57M07)},
mrnumber = {1253544},
zblnumber = {0841.20039},
} -
[GroCC] M. Gromov, "Carnot-Carathéodory spaces seen from within," in Sub-Riemannian Geometry, Birkhäuser, Basel, 1996, vol. 144, pp. 79-323.
@INCOLLECTION{GroCC,
author = {Gromov, Mikhael},
title = {Carnot-{C}arathéodory spaces seen from within},
booktitle = {Sub-{R}iemannian Geometry},
venue = {{P}aris, {F}rance, June 30--July 1, 1992},
series = {Progr. Math.},
volume = {144},
pages = {79--323},
publisher = {Birkhäuser, Basel},
year = {1996},
mrclass = {53C17 (53C23)},
mrnumber = {1421823},
mrreviewer = {Richard W. Montgomery},
zblnumber = {0864.53025},
} -
[Helg] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society, Providence, RI, 2001, vol. 34.
@BOOK{Helg,
author = {Helgason, Sigurdur},
title = {Differential Geometry, {L}ie Groups, and Symmetric Spaces},
series = {Grad. Stud. in Math.},
volume = {34},
note = {corrected reprint of the 1978 original},
publisher = {American Mathematical Society, Providence, RI},
year = {2001},
pages = {xxvi+641},
isbn = {0-8218-2848-7},
mrclass = {53C35 (22E10 22E46 22E60)},
mrnumber = {1834454},
doi = {10.1090/gsm/034},
url = {https://doi.org/10.1090/gsm/034},
zblnumber = {0993.53002},
} -
[KMLog] D. Y. Kleinbock and G. A. Margulis, "Logarithm laws for flows on homogeneous spaces," Invent. Math., vol. 138, iss. 3, pp. 451-494, 1999.
@ARTICLE{KMLog,
author = {Kleinbock, D. Y. and Margulis, G. A.},
title = {Logarithm laws for flows on homogeneous spaces},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {138},
year = {1999},
number = {3},
pages = {451--494},
issn = {0020-9910},
mrclass = {37C85 (11J99 22E40 37A30 37D30 37D40)},
mrnumber = {1719827},
mrreviewer = {S. G. Dani},
doi = {10.1007/s002220050350},
url = {https://doi.org/10.1007/s002220050350},
zblnumber = {0934.22016},
} -
[KMLogErrata] D. Y. Kleinbock and G. A. Margulis, "Erratum to: Logarithm laws for flows on homogeneous spaces," Invent. Math., vol. 211, iss. 2, pp. 855-862, 2018.
@ARTICLE{KMLogErrata,
author = {Kleinbock, D. Y. and Margulis, G. A.},
title = {Erratum to: {L}ogarithm laws for flows on homogeneous spaces},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {211},
year = {2018},
number = {2},
pages = {855--862},
issn = {0020-9910},
mrclass = {37C85 (11J99 22E40 37A30 37D30 37D40)},
mrnumber = {3748318},
doi = {10.1007/s00222-017-0751-3},
url = {https://doi.org/10.1007/s00222-017-0751-3},
zblnumber = {1404.22021},
} -
[LeuExh] E. Leuzinger, "An exhaustion of locally symmetric spaces by compact submanifolds with corners," Invent. Math., vol. 121, iss. 2, pp. 389-410, 1995.
@ARTICLE{LeuExh,
author = {Leuzinger, Enrico},
title = {An exhaustion of locally symmetric spaces by compact submanifolds with corners},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {121},
year = {1995},
number = {2},
pages = {389--410},
issn = {0020-9910},
mrclass = {53C35},
mrnumber = {1346213},
doi = {10.1007/BF01884305},
url = {https://doi.org/10.1007/BF01884305},
zblnumber = {0844.53040},
} -
[LeuCorank] E. Leuzinger, "Corank and asymptotic filling-invariants for symmetric spaces," Geom. Funct. Anal., vol. 10, iss. 4, pp. 863-873, 2000.
@ARTICLE{LeuCorank,
author = {Leuzinger, Enrico},
title = {Corank and asymptotic filling-invariants for symmetric spaces},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {10},
year = {2000},
number = {4},
pages = {863--873},
issn = {1016-443X},
mrclass = {53C23 (20F67 20F69 53C35)},
mrnumber = {1791143},
mrreviewer = {Michel Coornaert},
doi = {10.1007/PL00001641},
url = {https://doi.org/10.1007/PL00001641},
zblnumber = {1037.53038},
} -
[Leu-Kazh] E. Leuzinger, "Kazhdan’s property (T), $L^2$-spectrum and isoperimetric inequalities for locally symmetric spaces," Comment. Math. Helv., vol. 78, iss. 1, pp. 116-133, 2003.
@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},
} -
[LeuPoly] E. Leuzinger, "On polyhedral retracts and compactifications of locally symmetric spaces," Differential Geom. Appl., vol. 20, iss. 3, pp. 293-318, 2004.
@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},
} -
[LeuISO] E. Leuzinger, "Optimal higher-dimensional Dehn functions for some $ CAT(0)$ lattices," Groups Geom. Dyn., vol. 8, iss. 2, pp. 441-466, 2014.
@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},
} -
[LMR] A. Lubotzky, S. Mozes, and M. S. Raghunathan, "The word and riemannian metrics on lattices of semisimple groups," Inst. Hautes Études Sci. Publ. Math., vol. 91, pp. 5-53, 2000.
@ARTICLE{LMR,
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.},
fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
volume = {91},
year = {2000},
pages = {5--53},
issn = {0073-8301},
mrclass = {22E40 (53C23)},
mrnumber = {1828742},
mrreviewer = {B. Sury},
url = {http://www.numdam.org/item?id=PMIHES_2000__91__5_0},
zblnumber = {0988.22007},
} -
[LeuzPitRk2] E. Leuzinger and C. Pittet, "Isoperimetric inequalities for lattices in semisimple Lie groups of rank $2$," Geom. Funct. Anal., vol. 6, iss. 3, pp. 489-511, 1996.
@ARTICLE{LeuzPitRk2,
author = {Leuzinger, Enrico and Pittet, C.},
title = {Isoperimetric inequalities for lattices in semisimple {L}ie groups of rank {$2$}},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {6},
year = {1996},
number = {3},
pages = {489--511},
issn = {1016-443X},
mrclass = {22E40 (22E46 53C35 57M07)},
mrnumber = {1392327},
mrreviewer = {Raul Quiroga-Barranco},
doi = {10.1007/BF02249261},
url = {https://doi.org/10.1007/BF02249261},
zblnumber = {0856.22013},
} -
[LangSch] U. Lang and T. Schlichenmaier, "Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions," Int. Math. Res. Not., iss. 58, pp. 3625-3655, 2005.
@ARTICLE{LangSch,
author = {Lang, Urs and Schlichenmaier, Thilo},
title = {Nagata dimension, quasisymmetric embeddings, and {L}ipschitz extensions},
journal = {Int. Math. Res. Not.},
fjournal = {International Mathematics Research Notices},
year = {2005},
number = {58},
pages = {3625--3655},
issn = {1073-7928},
mrclass = {53C23 (30C65 53C21 54F45)},
mrnumber = {2200122},
mrreviewer = {Jeremy T. Tyson},
doi = {10.1155/IMRN.2005.3625},
url = {https://doi.org/10.1155/IMRN.2005.3625},
zblnumber = {1095.53033},
} -
[LeYoRank1] E. Leuzinger and R. Young, "The distortion dimension of $\Bbb Q$-rank 1 lattices," Geom. Dedicata, vol. 187, pp. 69-87, 2017.
@ARTICLE{LeYoRank1,
author = {Leuzinger, Enrico and Young, Robert},
title = {The distortion dimension of {$\Bbb Q$}-rank 1 lattices},
journal = {Geom. Dedicata},
fjournal = {Geometriae Dedicata},
volume = {187},
year = {2017},
pages = {69--87},
issn = {0046-5755},
mrclass = {22E40 (20F65 53C35)},
mrnumber = {3622683},
mrreviewer = {O. V. Shvartsman},
doi = {10.1007/s10711-016-0189-6},
url = {https://doi.org/10.1007/s10711-016-0189-6},
zblnumber = {1366.22006},
} -
[MoLOCSYM] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo, 1973, vol. 78.
@BOOK{MoLOCSYM,
author = {Mostow, G. D.},
title = {Strong Rigidity of Locally Symmetric Spaces},
series = {Ann. of Math. Stud.},
volume = {78},
publisher = {Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo},
year = {1973},
pages = {v+195},
mrclass = {22E40 (53C35)},
mrnumber = {0385004},
mrreviewer = {M. S. Raghunathan},
doi = {10.1515/9781400881833},
url = {https://doi.org/10.1515/9781400881833},
zblnumber = {0265.53039},
} -
[OhUniform] H. Oh, "Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants," Duke Math. J., vol. 113, iss. 1, pp. 133-192, 2002.
@ARTICLE{OhUniform,
author = {Oh, Hee},
title = {Uniform pointwise bounds for matrix coefficients of unitary representations and applications to {K}azhdan constants},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {113},
year = {2002},
number = {1},
pages = {133--192},
issn = {0012-7094},
mrclass = {22E46 (22E50)},
mrnumber = {1905394},
mrreviewer = {Mark R. Sepanski},
doi = {10.1215/S0012-7094-02-11314-3},
url = {https://doi.org/10.1215/S0012-7094-02-11314-3},
zblnumber = {1011.22007},
} -
[Warner] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups. I, Springer-Verlag, New York, 1972, vol. 188.
@BOOK{Warner,
author = {Warner, Garth},
title = {Harmonic Analysis on Semi-Simple {L}ie Groups. {I}},
note = {Grundlehren math. Wissen.},
volume = {188},
publisher = {Springer-Verlag, New York},
year = {1972},
pages = {xvi+529},
mrclass = {22E45 (22E30)},
mrnumber = {0498999},
mrreviewer = {Mitsuo Sugiura},
doi = {10.1007/978-3-642-50275-0},
url = {https://doi.org/10.1007/978-3-642-50275-0},
zblnumber = {0265.22020},
} -
[WengerShort] S. Wenger, "A short proof of Gromov’s filling inequality," Proc. Amer. Math. Soc., vol. 136, iss. 8, pp. 2937-2941, 2008.
@ARTICLE{WengerShort,
author = {Wenger, Stefan},
title = {A short proof of {G}romov's filling inequality},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {136},
year = {2008},
number = {8},
pages = {2937--2941},
issn = {0002-9939},
mrclass = {53C23},
mrnumber = {2399061},
mrreviewer = {Stéphane Sabourau},
doi = {10.1090/S0002-9939-08-09203-4},
url = {https://doi.org/10.1090/S0002-9939-08-09203-4},
zblnumber = {1148.53031},
} -
[WhiteDeform] B. White, "The deformation theorem for flat chains," Acta Math., vol. 183, iss. 2, pp. 255-271, 1999.
@ARTICLE{WhiteDeform,
author = {White, Brian},
title = {The deformation theorem for flat chains},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {183},
year = {1999},
number = {2},
pages = {255--271},
issn = {0001-5962},
mrclass = {49Q20 (49J45)},
mrnumber = {1738045},
mrreviewer = {Giovanni Bellettini},
doi = {10.1007/BF02392829},
url = {https://doi.org/10.1007/BF02392829},
zblnumber = {0980.49035},
} -
[WortExpLower] K. Wortman, "Exponential higher dimensional isoperimetric inequalities for some arithmetic groups," Geom. Dedicata, vol. 151, pp. 141-153, 2011.
@ARTICLE{WortExpLower,
author = {Wortman, Kevin},
title = {Exponential higher dimensional isoperimetric inequalities for some arithmetic groups},
journal = {Geom. Dedicata},
fjournal = {Geometriae Dedicata},
volume = {151},
year = {2011},
pages = {141--153},
issn = {0046-5755},
mrclass = {22E40 (20G30)},
mrnumber = {2780742},
mrreviewer = {Enrico Leuzinger},
doi = {10.1007/s10711-010-9523-6},
url = {https://doi.org/10.1007/s10711-010-9523-6},
zblnumber = {1219.20027},
} -
[YoungQuad] R. Young, "The Dehn function of ${ SL}(n;\Bbb Z)$," Ann. of Math. (2), vol. 177, iss. 3, pp. 969-1027, 2013.
@ARTICLE{YoungQuad,
author = {Young, Robert},
title = {The {D}ehn function of {${\rm SL}(n;\Bbb Z)$}},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {177},
year = {2013},
number = {3},
pages = {969--1027},
issn = {0003-486X},
mrclass = {20F65},
mrnumber = {3034292},
mrreviewer = {Mark Kambites},
doi = {10.4007/annals.2013.177.3.4},
url = {https://doi.org/10.4007/annals.2013.177.3.4},
zblnumber = {1276.20053},
} -
[YoungHigherSol] R. Young, "Lipschitz connectivity and filling invariants in solvable groups and buildings," Geom. Topol., vol. 18, iss. 4, pp. 2375-2417, 2014.
@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},
pages = {2375--2417},
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},
}
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