The Dehn function of $\mathrm{SL}(n;\mathbb{Z})$

Robert Young

doi:https://doi.org/10.4007/annals.2013.177.3.4

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.

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[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.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ A. Borel and Harish-Chandra, "Arithmetic subgroups of algebraic groups," Ann. of Math., vol. 75, pp. 485-535, 1962.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ K. Bux and K. Wortman, "Finiteness properties of arithmetic groups over function fields," Invent. Math., vol. 167, iss. 2, pp. 355-378, 2007.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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] $Go to document$ J. T. Ding, "A proof of a conjecture of C. L. Siegel," J. Number Theory, vol. 46, iss. 1, pp. 1-11, 1994.

Show bibtex

@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] $Go to document$ C. Dructu, "Filling in solvable groups and in lattices in semisimple groups," Topology, vol. 43, iss. 5, pp. 983-1033, 2004.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ M. Gromov, "Carnot-Carathéodory spaces seen from within," in Sub-Riemannian Geometry, Basel: Birkhäuser, 1996, vol. 144, pp. 79-323.

Show bibtex

@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] $Go to document$ L. Ji, "Metric compactifications of locally symmetric spaces," Internat. J. Math., vol. 9, iss. 4, pp. 465-491, 1998.

Show bibtex

@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] $Go to document$ L. Ji and R. MacPherson, "Geometry of compactifications of locally symmetric spaces," Ann. Inst. Fourier $($Grenoble$)$, vol. 52, iss. 2, pp. 457-559, 2002.

Show bibtex

@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] $Go to document$ E. Leuzinger, "On polyhedral retracts and compactifications of locally symmetric spaces," Differential Geom. Appl., vol. 20, iss. 3, pp. 293-318, 2004.

Show bibtex

@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.

Show bibtex

@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] $Go to document$ 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.

Show bibtex

@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] $Go to document$ E. Leuzinger and C. Pittet, "On quadratic Dehn functions," Math. Z., vol. 248, iss. 4, pp. 725-755, 2004.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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}, }
[RileyNav] $Go to document$ T. R. Riley, "Navigating in the Cayley graphs of ${ SL}_N(\Bbb Z)$ and ${ SL}_N(\Bbb F_p)$," Geom. Dedicata, vol. 113, pp. 215-229, 2005.

Show bibtex

@article {RileyNav, MRKEY = {2171306}, AUTHOR = {Riley, T. R.}, TITLE = {Navigating in the {C}ayley graphs of {${\rm SL}\sb N(\Bbb Z)$} and {${\rm SL}\sb N(\Bbb F\sb p)$}}, JOURNAL = {Geom. Dedicata}, FJOURNAL = {Geometriae Dedicata}, VOLUME = {113}, YEAR = {2005}, PAGES = {215--229}, ISSN = {0046-5755}, CODEN = {GEMDAT}, MRCLASS = {20F05}, MRNUMBER = {2171306}, MRREVIEWER = {Thomas Weigel}, DOI = {10.1007/s10711-005-5230-0}, ZBLNUMBER = {1110.20026}, }
[Steinberg] R. Steinberg, "Générateurs, relations et revêtements de groupes algébriques," in Colloq. Théorie des Groupes Algébriques, Librairie Universitaire, Louvain, 1962, pp. 113-127.

Show bibtex

@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).

Show bibtex

@misc{YoungQuart, author={Young, R.}, TITLE={A polynomial isoperimetric inequality for {SL(n,Z)}}, ARXIV= {0903.2495}, }

The Dehn function of $\mathrm{SL}(n;\mathbb{Z})$

Abstract

Authors