The Dehn function of \(\text{SL}(n;\mathbb Z)\). (English) Zbl 1276.20053

Ann. Math. (2) 177, No. 3, 969-1027 (2013); erratum 178, No. 3, 1199 (2013).
The Dehn function of a group equipped with a generating set is a geometric invariant that measures the difficulty in reducing a word written in the generators that represents the identity to the trivial word. In the paper under review, the author proves that for \(n\geq 5\), the Dehn function of \(\mathrm{SL}(n,\mathbb Z)\) is quadratic. Thurston conjectured that this holds for all \(n\geq 4\). The result for \(n=4\) is still open.
The proof reduces the probem of filling of words in \(\mathrm{SL}(n,\mathbb Z)\) to the problem of filling curves in the symmetric space \(\mathrm{SL}(n,\mathbb R)/\mathrm{SO}(n)\). The author’s proof involves then the decomposition of a disk 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”, the author obtains words of a particular form and he uses combinatorial techniques to fill in these loops. In particular, the arguments involve the question of filling in words in parabolic subgroups of the group.


20F65 Geometric group theory
57S25 Groups acting on specific manifolds
20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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[1] H. Behr, ”\({ SL}_3({\mathbf F}_q[t])\) is not finitely presentable,” in Homological Group Theory, Cambridge: Cambridge Univ. Press, 1979, pp. 213-224. · Zbl 0434.20025
[2] M. Bestvina, A. Eskin, and K. Wortman, Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups. · Zbl 1372.11055
[3] A. Borel and Harish-Chandra, ”Arithmetic subgroups of algebraic groups,” Ann. of Math., vol. 75, pp. 485-535, 1962. · Zbl 0107.14804
[4] 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. · Zbl 0996.54507
[5] K. Bux and K. Wortman, ”Finiteness properties of arithmetic groups over function fields,” Invent. Math., vol. 167, iss. 2, pp. 355-378, 2007. · Zbl 1126.20030
[6] 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. · Zbl 1254.20035
[7] J. T. Ding, ”A proof of a conjecture of C. L. Siegel,” J. Number Theory, vol. 46, iss. 1, pp. 1-11, 1994. · Zbl 0797.11058
[8] C. Dructu, ”Filling in solvable groups and in lattices in semisimple groups,” Topology, vol. 43, iss. 5, pp. 983-1033, 2004. · Zbl 1083.20033
[9] 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. · Zbl 0764.20017
[10] 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. · Zbl 0829.20054
[11] C. Groft, Generalized Dehn functions I.
[12] M. Gromov, ”Asymptotic invariants of infinite groups,” in Geometric Group Theory, Vol. 2, Cambridge: Cambridge Univ. Press, 1993, vol. 182, pp. 1-295. · Zbl 0841.20039
[13] M. Gromov, ”Carnot-Carathéodory spaces seen from within,” in Sub-Riemannian Geometry, Basel: Birkhäuser, 1996, vol. 144, pp. 79-323. · Zbl 0864.53025
[14] L. Ji, ”Metric compactifications of locally symmetric spaces,” Internat. J. Math., vol. 9, iss. 4, pp. 465-491, 1998. · Zbl 0929.32017
[15] L. Ji and R. MacPherson, ”Geometry of compactifications of locally symmetric spaces,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 52, iss. 2, pp. 457-559, 2002. · Zbl 1017.53039
[16] E. Leuzinger, ”On polyhedral retracts and compactifications of locally symmetric spaces,” Differential Geom. Appl., vol. 20, iss. 3, pp. 293-318, 2004. · Zbl 1052.22008
[17] 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. · Zbl 1086.53073
[18] 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. · Zbl 0856.22013
[19] E. Leuzinger and C. Pittet, ”On quadratic Dehn functions,” Math. Z., vol. 248, iss. 4, pp. 725-755, 2004. · Zbl 1132.20021
[20] 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. · Zbl 0786.22016
[21] J. Milnor, Introduction to Algebraic \(K\)-Theory, Princeton, N.J.: Princeton Univ. Press, 1971, vol. 72. · Zbl 0237.18005
[22] T. R. Riley, ”Navigating in the Cayley graphs of \({ SL}_N(\mathbb Z)\) and \({ SL}_N(\mathbb F_p)\),” Geom. Dedicata, vol. 113, pp. 215-229, 2005. · Zbl 1110.20026
[23] R. Steinberg, ”Générateurs, relations et revêtements de groupes algébriques,” in Colloq. Théorie des Groupes Algébriques, Librairie Universitaire, Louvain, 1962, pp. 113-127. · Zbl 0272.20036
[24] R. Young, A polynomial isoperimetric inequality for SL(n,Z).
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