Abstract
We prove that when $n\ge 5$, the Dehn function of $\mathrm{SL}(n;\mathbb{Z})$ is quadratic. The proof involves decomposing a disc in $\mathrm{SL}(n;\mathbb{R})/\mathrm{SO}(n)$ into triangles of varying sizes. By mapping these triangles into $\mathrm{SL}(n;\mathbb{Z})$ and replacing large elementary matrices by “shortcuts,” we obtain words of a particular form, and we use combinatorial techniques to fill these loops.
-
[Behr77] H. Behr, "${ SL}_{3}({\bf F}_{q}[t])$ is not finitely presentable," in Homological Group Theory, Cambridge: Cambridge Univ. Press, 1979, pp. 213-224.
@incollection {Behr77, MRKEY = {0564424},
AUTHOR = {Behr, Helmut},
TITLE = {{${\rm SL}\sb{3}({\bf F}\sb{q}[t])$} is not finitely presentable},
BOOKTITLE = {Homological Group Theory},
VENUE={{P}roc. {S}ympos., {D}urham, 1977},
SERIES = {London Math. Soc. Lecture Note Ser.},
NUMBER = {36},
PAGES = {213--224},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1979},
MRCLASS = {20G30 (20F05)},
MRNUMBER = {0564424},
MRREVIEWER = {Ulf Rehmann},
ZBLNUMBER = {0434.20025},
} -
[BestEskWort] M. Bestvina, A. Eskin, and K. Wortman, Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups.
@misc{BestEskWort,
author={Bestvina, M. and Eskin, A. and Wortman, K.},
TITLE={Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups},
ARXIV={1106.0162},
} -
[BorHar-Cha]
A. Borel and Harish-Chandra, "Arithmetic subgroups of algebraic groups," Ann. of Math., vol. 75, pp. 485-535, 1962.
@article {BorHar-Cha, MRKEY = {0147566},
AUTHOR = {Borel, Armand and Harish-Chandra},
TITLE = {Arithmetic subgroups of algebraic groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {75},
YEAR = {1962},
PAGES = {485--535},
ISSN = {0003-486X},
MRCLASS = {20.65 (14.50)},
MRNUMBER = {0147566},
MRREVIEWER = {P. Cartier},
ZBLNUMBER = {0107.14804},
DOI = {10.2307/1970210},
} -
[Bridson] M. R. Bridson, "The geometry of the word problem," in Invitations to Geometry and Topology, Oxford: Oxford Univ. Press, 2002, vol. 7, pp. 29-91.
@incollection {Bridson, MRKEY = {1967746},
AUTHOR = {Bridson, Martin R.},
TITLE = {The geometry of the word problem},
BOOKTITLE = {Invitations to Geometry and Topology},
SERIES = {Oxf. Grad. Texts Math.},
VOLUME = {7},
PAGES = {29--91},
PUBLISHER = {Oxford Univ. Press},
ADDRESS = {Oxford},
YEAR = {2002},
MRCLASS = {20F65 (20F10 49J05 57M07)},
MRNUMBER = {1967746},
MRREVIEWER = {Ross Geoghegan},
ZBLNUMBER = {0996.54507},
} -
[BuxWortman]
K. Bux and K. Wortman, "Finiteness properties of arithmetic groups over function fields," Invent. Math., vol. 167, iss. 2, pp. 355-378, 2007.
@article {BuxWortman, MRKEY = {2270455},
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},
CODEN = {INVMBH},
MRCLASS = {11F75 (11F06 20G35 22E40 57M07)},
MRNUMBER = {2270455},
MRREVIEWER = {Herbert Abels},
DOI = {10.1007/s00222-006-0017-y},
ZBLNUMBER = {1126.20030},
} -
[deCorTess]
Y. de Cornulier and R. Tessera, "Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups," Confluentes Math., vol. 2, iss. 4, pp. 431-443, 2010.
@article {deCorTess, MRKEY = {2764812},
AUTHOR = {de Cornulier, Yves and Tessera, Romain},
TITLE = {Metabelian groups with quadratic {D}ehn function and {B}aumslag-{S}olitar groups},
JOURNAL = {Confluentes Math.},
FJOURNAL = {Confluentes Mathematici},
VOLUME = {2},
YEAR = {2010},
NUMBER = {4},
PAGES = {431--443},
ISSN = {1793-7442},
MRCLASS = {20F65 (20F16 20F69)},
MRNUMBER = {2764812},
MRREVIEWER = {Josh B. Barnard},
DOI = {10.1142/S1793744210000235},
ZBLNUMBER = {05849551},
} -
[Ding]
J. T. Ding, "A proof of a conjecture of C. L. Siegel," J. Number Theory, vol. 46, iss. 1, pp. 1-11, 1994.
@article {Ding, MRKEY = {1268360},
AUTHOR = {Ding, Jin Tai},
TITLE = {A proof of a conjecture of {C}. {L}. {S}iegel},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {46},
YEAR = {1994},
NUMBER = {1},
PAGES = {1--11},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11H55},
MRNUMBER = {1268360},
MRREVIEWER = {Jeffrey Lin Thunder},
DOI = {10.1006/jnth.1994.1001},
ZBLNUMBER = {0797.11058},
} -
[DrutuFilling]
C. Dructu, "Filling in solvable groups and in lattices in semisimple groups," Topology, vol. 43, iss. 5, pp. 983-1033, 2004.
@article{DrutuFilling, MRKEY = {2079992},
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},
CODEN = {TPLGAF},
MRCLASS = {20F10 (20F65 22E25 22E40 51E24)},
MRNUMBER = {2079992},
MRREVIEWER = {Vitaly A. Roman{\cprime}kov},
DOI = {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, Boston, MA: Jones and Bartlett Publishers, 1992.
@book {ECHLPT, MRKEY={1161694},
AUTHOR={Epstein, D. B. A. and Cannon, J. W. and Holt, D. F. and Levy, S. V. F. and Paterson, M. S. and Thurston, W. P.},
TITLE={Word Processing in Groups},
PUBLISHER={Jones and Bartlett Publishers},
ADDRESS={Boston, MA},
YEAR={1992},
MRNUMBER={1161694},
ZBLNUMBER = {0764.20017},
} -
[GerstenSurv]
S. M. Gersten, "Isoperimetric and isodiametric functions of finite presentations," in Geometric Group Theory, Vol. 1, Cambridge: Cambridge Univ. Press, 1993, vol. 181, pp. 79-96.
@incollection {GerstenSurv, MRKEY = {1238517},
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},
ADDRESS = {Cambridge},
YEAR = {1993},
MRCLASS = {20F06 (57M07)},
MRNUMBER = {1238517},
MRREVIEWER = {John Meier},
DOI = {10.1017/CBO9780511661860.008},
ZBLNUMBER = {0829.20054},
} -
[GroftA] C. Groft, Generalized Dehn functions I.
@misc{GroftA,
author={Groft, C.},
TITLE={Generalized {D}ehn functions {I}},
ARXIV={0901.2303},
} -
[GroAII] M. Gromov, "Asymptotic invariants of infinite groups," in Geometric Group Theory, Vol. 2, Cambridge: Cambridge Univ. Press, 1993, vol. 182, pp. 1-295.
@incollection {GroAII, MRKEY = {1253544},
AUTHOR = {Gromov, M.},
TITLE = {Asymptotic invariants of infinite groups},
BOOKTITLE = {Geometric Group Theory, {V}ol. 2},
VENUE={{S}ussex, 1991},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {182},
PAGES = {1--295},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1993},
MRCLASS = {20F32 (57M07)},
MRNUMBER = {1253544},
ZBLNUMBER = {0841.20039},
} -
[GroCC]
M. Gromov, "Carnot-Carathéodory spaces seen from within," in Sub-Riemannian Geometry, Basel: Birkhäuser, 1996, vol. 144, pp. 79-323.
@incollection {GroCC, MRKEY = {1421823},
AUTHOR = {Gromov, Mikhael},
TITLE = {Carnot-{C}arathéodory spaces seen from within},
BOOKTITLE = {Sub-{R}iemannian Geometry},
SERIES = {Progr. Math.},
VOLUME = {144},
PAGES = {79--323},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {1996},
MRCLASS = {53C17 (53C23)},
MRNUMBER = {1421823},
MRREVIEWER = {Richard Montgomery},
ZBLNUMBER = {0864.53025},
DOI = {10.1007/978-3-0348-9210-0_2},
} -
[Ji]
L. Ji, "Metric compactifications of locally symmetric spaces," Internat. J. Math., vol. 9, iss. 4, pp. 465-491, 1998.
@article {Ji, MRKEY = {1635185},
AUTHOR = {Ji, Lizhen},
TITLE = {Metric compactifications of locally symmetric spaces},
JOURNAL = {Internat. J. Math.},
FJOURNAL = {International Journal of Mathematics},
VOLUME = {9},
YEAR = {1998},
NUMBER = {4},
PAGES = {465--491},
ISSN = {0129-167X},
MRCLASS = {32J05 (32H20)},
MRNUMBER = {1635185},
MRREVIEWER = {Raul Quiroga},
DOI = {10.1142/S0129167X98000208},
ZBLNUMBER = {0929.32017},
} -
[JiMacPherson]
L. Ji and R. MacPherson, "Geometry of compactifications of locally symmetric spaces," Ann. Inst. Fourier $($Grenoble$)$, vol. 52, iss. 2, pp. 457-559, 2002.
@article {JiMacPherson, MRKEY = {1906482},
AUTHOR = {Ji, L. and MacPherson, R.},
TITLE = {Geometry of compactifications of locally symmetric spaces},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {52},
YEAR = {2002},
NUMBER = {2},
PAGES = {457--559},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {22E40 (20G30 31C35 53C35 54D35)},
MRNUMBER = {1906482},
MRREVIEWER = {Enrico Leuzinger},
URL = {http://aif.cedram.org/item?id=AIF_2002__52_2_457_0},
ZBLNUMBER = {1017.53039},
} -
[LeuzingerPolyRet]
E. Leuzinger, "On polyhedral retracts and compactifications of locally symmetric spaces," Differential Geom. Appl., vol. 20, iss. 3, pp. 293-318, 2004.
@article {LeuzingerPolyRet, MRKEY = {2053916},
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},
CODEN = {DGAPEO},
MRCLASS = {53C35 (22E40 53C20)},
MRNUMBER = {2053916},
DOI = {10.1016/j.difgeo.2004.03.001},
ZBLNUMBER = {1052.22008},
} -
[Leuzinger] E. Leuzinger, "Tits geometry, arithmetic groups, and the proof of a conjecture of Siegel," J. Lie Theory, vol. 14, iss. 2, pp. 317-338, 2004.
@article {Leuzinger, MRKEY = {2066859},
AUTHOR = {Leuzinger, Enrico},
TITLE = {Tits geometry, arithmetic groups, and the proof of a conjecture of {S}iegel},
JOURNAL = {J. Lie Theory},
FJOURNAL = {Journal of Lie Theory},
VOLUME = {14},
YEAR = {2004},
NUMBER = {2},
PAGES = {317--338},
ISSN = {0949-5932},
MRCLASS = {53C24 (22E40)},
MRNUMBER = {2066859},
MRREVIEWER = {Dmitry Y. Kleinbock},
ZBLNUMBER = {1086.53073},
} -
[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, MRKEY = {1392327},
AUTHOR = {Leuzinger, E. 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},
CODEN = {GFANFB},
MRCLASS = {22E40 (22E46 53C35 57M07)},
MRNUMBER = {1392327},
MRREVIEWER = {Raul Quiroga},
DOI = {10.1007/BF02249261},
ZBLNUMBER = {0856.22013},
} -
[LeuzPitQuad]
E. Leuzinger and C. Pittet, "On quadratic Dehn functions," Math. Z., vol. 248, iss. 4, pp. 725-755, 2004.
@article {LeuzPitQuad, MRKEY = {2103539},
AUTHOR = {Leuzinger, E. and Pittet, Ch.},
TITLE = {On quadratic {D}ehn functions},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {248},
YEAR = {2004},
NUMBER = {4},
PAGES = {725--755},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {20F65 (20F69 53C35)},
MRNUMBER = {2103539},
MRREVIEWER = {Stephen G. Brick},
DOI = {10.1007/s00209-004-0678-4},
ZBLNUMBER = {1132.20021},
} -
[LMRComptes] A. Lubotzky, S. Mozes, and M. S. Raghunathan, "Cyclic subgroups of exponential growth and metrics on discrete groups," C. R. Acad. Sci. Paris Sér. I Math., vol. 317, iss. 8, pp. 735-740, 1993.
@article {LMRComptes, MRKEY = {1244421},
AUTHOR = {Lubotzky, Alexander and Mozes, Shahar and Raghunathan, M. S.},
TITLE = {Cyclic subgroups of exponential growth and metrics on discrete groups},
JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
VOLUME = {317},
YEAR = {1993},
NUMBER = {8},
PAGES = {735--740},
ISSN = {0764-4442},
CODEN = {CASMEI},
MRCLASS = {22E40 (53C23)},
MRNUMBER = {1244421},
MRREVIEWER = {Nimish A. Shah},
ZBLNUMBER = {0786.22016},
} -
[Milnor] J. Milnor, Introduction to Algebraic $K$-Theory, Princeton, N.J.: Princeton Univ. Press, 1971, vol. 72.
@book {Milnor, MRKEY = {0349811},
AUTHOR = {Milnor, John},
TITLE = {Introduction to Algebraic {$K$}-Theory},
SERIES = {Ann. of Math. Studies},
VOLUME={72},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1971},
PAGES = {xiii+184},
MRCLASS = {18F25 (12A65)},
MRNUMBER = {0349811},
MRREVIEWER = {J.-P. Jouanolou},
ZBLNUMBER = {0237.18005},
} -
[Steinberg] R. Steinberg, "Générateurs, relations et revêtements de groupes algébriques," in Colloq. Théorie des Groupes Algébriques, Librairie Universitaire, Louvain, 1962, pp. 113-127.
@incollection {Steinberg, MRKEY = {0153677},
AUTHOR = {Steinberg, Robert},
TITLE = {Générateurs, relations et revêtements de groupes algébriques},
BOOKTITLE = {Colloq. {T}héorie des {G}roupes {A}lgébriques},
VENUE={{B}ruxelles, 1962},
PAGES = {113--127},
PUBLISHER = {Librairie Universitaire, Louvain},
YEAR = {1962},
MRCLASS = {14.50 (20.75)},
MRNUMBER = {0153677},
MRREVIEWER = {T. Ono},
ZBLNUMBER = {0272.20036},
} -
[YoungQuart] R. Young, A polynomial isoperimetric inequality for SL(n,Z).
@misc{YoungQuart,
author={Young, R.},
TITLE={A polynomial isoperimetric inequality for {SL(n,Z)}},
ARXIV= {0903.2495},
}