The Isoperimetric Inequality for SL(n,Z)
This page concerns a workshop held at the American Institute of Mathematics, September 812, 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?
Background
Isoperimetric inequalities
There are two major viewpoints:
 Combinatorial.
An isoperimetric inequality for a finitely presented group is a measure
of the complexity of its word problem. It concerns proving that words
represent the identity by direct applications of the defining
relations.
 Geometric.
An isoperimetric inequality records the minimal areas of discs spanning
loops in a space. The classical example is the quadratic isoperimeric
inequality enjoyed by Euclidean space: loops of length L can be filled with discs of area at most a constant times the square of L.
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
illunderstood 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 proofstrategy outlined by Gromov [12, §5A_{7}] 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, noone 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 kspheres by
(k+1)discs agree with those for Euclidean space
for k ≤ n3.
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 "Rbuildings". 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 higherdimensional
version of the Lubotzky–Mozes–Raghunathan Theorem as the former can be
regarded as concerning finding efficient fillings of 1spheres by
2discs and latter efficient fillings of 0spheres by 1discs. 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 2skeleta 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 (n1)(n2)/2) in spines for SL(n,Z).
Supplementary goals of the workshop
 Enhance our understanding of SL(n,Z), and more generally of the geometry of many important groups (particularly noncocompact lattices in Lie groups).
 Many of the workshop participants are experts in various ways of looking at SL(n,Z) such as via its word metric, as a lattice (cocompact in neutered space), via buildings, and via number theory. We will seek to share and interrelate these perspectives.
 Develop new tools for establishing isoperimetric inequalities.
 Bring together and provoke interaction between experts on
combinatorial group theory, arithmetic groups, differential geometry,
and number theory.
 Stimulate progress on a number of related research directions.
Related open problems
A number of related open problems were raised before and during the workshop, including many in an OpenProblem Session led by Talia Fernós and recorded by Piotr Przytycki.

Determine the behaviour of other filling functions for SL(n,Z) such as its isodiametric and filling length functions. Do both admit linear upper bounds? Explore connections with the distortion of IA_{n} in Out(F_{n}).

Verify Gromov's suggestion that SL(n,Z) admits a Euclidean kth order Dehn function (concerning filling kspheres with (k+1)balls) for k < n2 — [12, §5D(5)(c)] (cf. §2B_{1}). Also show it is linear for all k > n2.
More generally, is it true that any nonuniform irreducible Rrank n lattice in a semisimple real Lie group admits a Euclidean kth order Dehn function for all k < n1, and is exponential for k = n1?
For results in this direction see

the exponential lower bound for the (n2)nd order Dehn function of SL(n,Z) established in [10],
 [7] on the Qrank one case,
 and [15], where an exponential firstorder Dehn function is proved for Rrank two.

Does every Qrank one lattice in SL(n,R) or in the product of at least three copies of SL(2,R) enjoy a quadratic isoperimetric function? — cf. [7]. Explicit examples to try are SL(2,Z[1/6]) and the Hilbert modular group SL(n,O), where O is the ring of integers of a totally real extension of Q of degree > 2.

Prove that SL(n,R)/SO(n) with two (or three, or four,... ) intersecting horoballs (appropriately defined) enjoys a quadratic isoperimetric function.
 Determine the connectedness properties of asymptotic cones of SL(n,Z) — [12, §2B_{1}].
Describe the horoballs one must remove from an asymptotic cone of SL(n,R)/SO(n) to get an asymptotic cone of SL(n,Z). Does every biLipschitz map of an asymototic cone of SL(n,Z) to itself arise from a quasiisometry of SL(n,Z)?
 Does SL(4,Z) enjoy Chatterji's L_{δ} property? — for background see [6] and [9]. This condition would imply a subcubic isoperimetric function and that the asymptotic cones are simply connnected.

Give a new elementary proof that SL(3,Z) admits an exponential isoperimetric inequality. [Gromov gave a strategy [12, §5A_{7}], which was subsequently fleshed out by E. Leuzinger — see Corollary 5.4 in [14].]
 What are the higher dimensional distortion functions of SL(n,Z) ("filling at infinity")? cf. [8]

What isoperimetric functions do Aut(F_{n}) and Out(F_{n}) enjoy? — [4], [24]. There is a standard loop (the "baseball curve") known to require an exponential area filling in SL(3,Z) — see [10]. It is known to lift to a loop in Out(F_{4}) and to have a quadraticarea filling in SL(4,Z). Does this filling lift to Out(F_{4})?
Similar questions can be asked about Sp(2n,Z). In particular one can investigate lifting words representing the identity in Sp(2n,Z) to words in the mapping class group so as to be able use the quadratic isoperimetric inequality of the mapping class group [20] to establish a quadratic isoperimetric inequality for Sp(2n,Z).
 Are there analogs for finitestate automata which lend themselves to questions of filling area (e.g. automata that can "see" δ–neighborhoods of 2disks)?
 Explicity (efficiently?) describe the geodesics in the Cayley graph of SL(n,Z) with respect to some set of generators.
 Is SL(3,Z) relatively combable as a subgroup of SL(4,Z)?
 What is the Dehn function of SL(n,Z[x]) for n > 3?
 Understand projection onto flats in SL(n,Z).
 For what k is SL(n,F_{q}[t]) kconnected at infinity? — one supposes k < n2.
 Do all loops in SL(3,Z) admit quadratic (or polynomial) area fillings in SL(4,Z)?
What about loops in SL(3,Z)semidirectZ^{3}?
In particular, what are the filling areas of the family of group constructed by Martin Kassabov (details to follow)?
 What is the abelianized (or homological) Dehn function of SL(n,Z)?
 Give a combinatorial proof of the Normal Subgroup Theorem for SL(n,Z).

Prove any nonquasiisometryinvariant statement on the word metric of SL(n,Z). For example, are balls in SL(n,Z) almost convex? Can there be dead ends of unbounded deapth? [The point of the nonquasiisometryinvariant condition is to hamper use of LubotzkyMozesRaghunathan.]
 Consider the following measure of hyperbolicity for a group with a word metric. Amongst all pairs of group elements x, y a distance r from the identity, take the average of d(x, y) and divide by r. [This should tend to 2 if G is a nonelementary hyperbolic group.]
Is this a quasiisometry invariant?
How does it behave in SL(n,Z)?



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