The Isoperimetric Inequality for SL(n,Z)

Update, June 2011
— R. Young, The Dehn function of SL(n;Z)
— M. Bestvina, A. Eskin and K. Wortman, Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

This page concerns a workshop held at the American Institute of Mathematics, September 8-12, 2008. Pages dedicated to the workshop on the AIM web site are here and here.

The workshop's focus was the question:

Does SL(n,Z) admit a quadratic isoperimetric inequality when n ≥ 4?

Group Photo


Isoperimetric inequalities

There are two major viewpoints:

If a group acts on a space suitably nicely, then any combinatorial isoperimetric inequality for the group is also (qualitatively) a geometric isoperimetric inequality for the space, and vice versa.

It is striking that isoperimetric inequalities remain ill-understood for many groups that are of fundamental interest in mathematics. The outstanding instance is SL(n,Z).

What is known about isoperimetric functions for SL(n,Z)?

SL(2,Z) has a linear isoperimetric inequality. In other words, SL(2,Z) is a hyperbolic group. Epstein and Thurston [10] proved that any isoperimetric inequality for SL(3,Z) grows at least exponentially quickly — that is, the number of relations required to reduce words w that represent the identity in SL(3,Z) to the empty word is sometimes at least exponential in the length of w. This is sharp: SL(n,Z) enjoys an exponential isoperimetric inequality for all n — a proof-strategy outlined by Gromov [12, §5A7] was taken up by Leuzinger [14, Cor. 5.4].


The origins of the problem

Thurston asserted (see [11]) that SL(4,Z) satisfies a quadratic isoperimetric inequality (the “smallest” inequality possible for a group that is not hyperbolic). But a proof was not forthcoming, and despite great interest from the field, no-one has been able to supply one. Indeed, when n ≥ 4, it is unknown whether SL(n,Z) admits a polynomial isoperimetric function.

Thurston's assertion was reinforced and extended by Gromov [12, §5D(5)(c)] when he suggested that in SL(n,Z), higher isoperimetric inequalities concerning filling k-spheres by (k+1)-discs agree with those for Euclidean space for k ≤ n-3.

Asymptotic cones

Asymptotic cones are limits of metric spaces viewed from a sequence of increasingly distant vantage points. Reasons for thinking they might be useful in the study of SL(n,Z) are that their topology is related to filling invariants in the original space, and the asymptotioc cones of SL(n,Z) have special structure: they are "R-buildings". Notes written by Robert Young giving some background on asymptotic cones are here (pdf).

Further background

Further background on the problem of determining the optimal isoperimetric inequality for SL(n,Z) can be found in [22] and [25].

Possible attacks

There are two prominent views on how to understand isoperimetric inequalities for SL(n,Z). They are markedly different as one is in the spirit of the combinatorial approach to isoperimetric inequalites, and the other the geometric approach.


The combintorial approach is to develop a qualitative version of a proof of finite presentability of SL(n,Z) (such as in [19]). One might use techniques such as are employed in proofs of the Lubotzky–Mozes–Raghunathan Theorem on word length in SL(n,Z) — see [17], [18], [21]. Indeed, Thurston's assertion may be regarded as a higher-dimensional version of the Lubotzky–Mozes–Raghunathan Theorem as the former can be regarded as concerning finding efficient fillings of 1-spheres by 2-discs and latter efficient fillings of 0-spheres by 1-discs. The fact that for n ≥ 3, SL(n,Z) does not admit a combing in which the combing lines have at most polynomial length [10] may make this approach difficult.

In the geometric approach, one considers SL(n,Z) as a lattice in the symmetric space X = SL(n,R) / SO(n). The difficultly is that this lattice is not cocompact, and so the quadratic isoperimetric inequality enjoyed by X does not immediately pass to SL(n,Z). The attempted remedy is to remove certain open horoballs from X to give a space ("neutered space") on which SL(n,Z) acts cocompactly. But the removed horoballs have complicated intersections (indeed, there are competing definitions for horoballs to sift through), and it is unclear whether their removal destroys the isoperiemtric inequality. Partial results in this direction can be found in [7] and [16].

Spines for SL(n,Z) relate to both the combinatorial and geometric approaches. A spine for SL(n,Z) is a cell complex usually embedded in a locally symmetric space which is still a classifying space for SL(n,Z). Number Theorists have investigated the combinatorics of spines (see [1] and [2]) as well as retractions of locally symmetric spaces onto them. The problem of finding the optimal isoperimetric inequality for SL(n,Z) may be viewed as a question about the relationship between the 1- and 2-skeleta of a spine. Results such as those in [3] demonstrate some of the techniques Number Theorists have used to understand the cells of various dimensions (in this case (n-1)(n-2)/2) in spines for SL(n,Z).

Supplementary goals of the workshop


Related open problems

A number of related open problems were raised before and during the workshop, including many in an Open-Problem Session led by Talia Fernós and recorded by Piotr Przytycki.

Talia_at_the_board Nate_Robert_Martin


  1. A. Ash, Deformation retracts with lowest possible dimension of arithmetic quotients of self-adjoint homogeneous cones, Math. Ann. 225 (1977), no. 1, 69–76

  2. A. Ash, Small-dimensional classifying spaces for arithmetic subgroups of general linear groups, Duke Math. J. 51 (1984), no. 2, 459–468

  3. A. Ash and L. Rudolph, The modular symbol and continued fractions in higher dimensions, Invent. Math. 55 (1979), no. 3, 241–250

  4. M. R. Bridson and K. Vogtmann. Automorphism groups of free, surface, and free abelian groups. In Problems on Mapping Class Groups, edited by B. Farb, Proceedings of Symposia in Pure Mathematics 74, American Mathematical Society, Providence, RI, 2006, pp. 301–316.

  5. M. R. Bridson and K. Vogtmann. On the geometry of the automorphism group of a free group. Bull. London Math. Soc., 27(6): 544–552, 1995

  6. I. Chatterji. On property (RD) for certain discrete groups. PhD Thesis, 2001, Zürich

  7. C. Drutu. Filling in solvable groups and in lattices in semisimple groups. Topology, 43:983–1033, 2004

  8. C. Drutu, S. Mozes, M. Sapir. Divergence in lattices in semisimple Lie groups and graphs of groups. 2008

  9. M. Elder. Lδ groups are almost convex and have sub-cubic Dehn function. Algebraic and Geometric Topology, Vol 4, 23–29, 2004.

  10. 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, 1992

  11. S. M. Gersten. Isoperimetric and isodiametric functions. In G. Niblo and M. Roller, editors, Geometric group theory I, number 181 in LMS lecture notes. Camb. Univ. Press, 1993

  12. M. Gromov. Asymptotic invariants of infinite groups.In G. Niblo and M. Roller, editors, Geometric group theory II, number 182 in LMS lecture notes. Camb. Univ. Press, 1993

  13. A. Hatcher and K. Vogtmann. Isoperimetric inequalities for automorphism groups of free groups. Pacific J. Math., 173(2): 425–441, 1996

  14. E. Leuzinger. On polyhedral retracts and compactifications of locally symmetric spaces. Diff Geom and its Appli., 20:293–318, 2004

  15. E. Leuzinger and Ch. Pittet. Isoperimetric inequalities for lattices in semisimple Lie groups of rank 2. Geom. Funct. Anal., 6(3):489–511, 1996

  16. E. Leuzinger and Ch. Pittet. On quadratic Dehn functions. Math. Z., 248(4):725–755, 2004

  17. A. Lubotzky, S. Mozes, and M. S. Raghunathan. Cyclic subgroups of exponential growth and metrics on discrete groups. C.R. Acad. Sci. Paris, Sèrie 1, 317:723–740, 1993

  18. A. Lubotzky, S. Mozes, and M. S. Raghunathan. The word and Riemannian metrics on lattices of semisimple groups. Inst. Hautes Études Sci. Publ. Math., 91:5–53, 2000

  19. J. Milnor. Introduction to algebraic K-theory, volume 72 of Annals of Mathematical Studies. Princeton University Press, 1971

  20. L. Mosher, Mapping class groups are automatic, Ann. of Math. (2) 142 (1995), no. 2, 303–384

  21. T. R. Riley. Navigating the Cayley graphs of SLN(Z) and SLN(Fp). Geometriae Dedicata, 113(1):215–229, 2005

  22. T. R. Riley. The Dehn function of SLn(Z), to appear in L'Enseignement Mathèmatique

  23. L. Saper, Tilings and finite energy retractions of locally symmetric spaces, Comment. Math. Helv. 72 (1997), no. 2, 167–202

  24. K. Vogtmann. Out(Fn), Problems in Geometric Group Theory wiki

  25. K. Wortman. Lattices in Lie Groups, Problems in Geometric Group Theory wiki