Lipschitz connectivity and filling invariants in solvable groups and buildings
We give some new methods, based on Lipschitz extension theorems, for bounding filling invariants of subsets of nonpositively curved spaces. We apply our methods to find sharp bounds on higher-order Dehn functions of , horospheres in euclidean buildings, Hilbert modular groups, and certain -arithmetic groups.
Filling invariants of a group or space, such as Dehn functions and higher-order Dehn functions, are quantitative versions of finiteness properties. There are many methods for bounding the Dehn function, but bounds on the Dehn function are often difficult to generalize to higher-order Dehn functions. For example, one can prove that a non-positively curved space has a Dehn function which is at most quadratic in a couple of lines: the fact that the distance function is convex implies that the disc formed by connecting every point on the curve to a basepoint on the curve has quadratically large area. On the other hand, proving that a non-positively curved space has a th-order Dehn function bounded by takes several pages [Gro83, Wen08]. In this paper, we describe some new methods for bounding higher-order Dehn functions and apply them to solvable groups and subsets of nonpositively curved spaces.
One reason that higher-order Dehn functions are harder to bound is that the geometry of spheres is more complicated than the geometry of curves. A closed curve is geometrically very simple. It has diameter bounded by its length, it has a natural parameterization by length, and a closed curve in a space with a geometric group action can be approximated by a word in the group. None of these hold for spheres. A -sphere of volume may have arbitrarily large diameter, has no natural parameterization, and, though it can often be approximated by a cellular or simplicial sphere, that sphere may have arbitrarily many cells of dimension less than .
One way around this is to consider Lipschitz extension properties. A typical Lipschitz extension property is Lipschitz -connectivity; we say that a space is Lipschitz -connected (with constant ) if there is a such that for any and any -Lipschitz map , there is a -Lipschitz extension . The advantage of dealing with Lipschitz spheres rather than spheres of bounded volume is that techniques for filling closed curves often generalize to Lipschitz spheres. For example, the same construction that shows that a non-positively curved space has quadratic Dehn function shows that such a space is Lipschitz -connected for any . Any map can be extended to a map by coning off to a point along geodesics, and if is Lipschitz, so is .
In this paper, we describe a way to use Lipschitz connectivity to prove bounds on higher-order filling functions of subsets of spaces with finite Assouad-Nagata dimension. These spaces include euclidean buildings and homogeneous Hadamard manifolds [LS05], and we will show that a higher-dimensional analogue of the Lubotzky-Mozes-Raghunathan theorem holds for Lipschitz -connected subsets of spaces with finite Assouad-Nagata dimension. Recall that Lubotzky, Mozes, and Raghunathan proved that
[LMR00] If is an irreducible lattice in a semisimple group of rank , then the word metric on is quasi-isometric to the restriction of the metric on to .
One way to state this theorem is that the inclusion does not induce any distortion of lengths. That is, there is a such that if are connected by a path of length in , then they are connected by a path of length in the Cayley graph of . We can think of this as an efficient 1-dimensional filling of a 0-sphere. Many authors have conjectured that when has higher rank, we can fill higher-dimensional spheres efficiently; for example, Thurston famously conjectured that has quadratic Dehn function for [Ger93], and Gromov conjectured that the -th order Dehn function of a lattice in a symmetric space of rank should be bounded by a polynomial [Gro93]. Bux and Wortman [BW07] conjectured that filling volumes should be undistorted in lattices in higher-rank semisimple groups. We will state a version of this conjecture in terms of homological filling volumes; in a highly-connected space, these are equivalent to homotopical filling volumes in dimensions above 2 [Gro83, Whi84, Gro].
To state the conjecture, we introduce Lipschitz chains. A Lipschitz -chain in is a formal sum of Lipschitz maps . One can define the boundary of a Lipschitz chain as for singular chains, and this gives rise to a homology theory. If is a Lipschitz -cycle in , define
to be the filling volume of in . In particular, if is a geodesic metric space and is the 0-cycle , then .
If , we say that is undistorted up to dimension if there is some and such that if is a Lipschitz -cycle in and , then
(Note that this differs from Bux and Wortman’s definition in [BW07]; Bux and Wortman’s definition deals with extending spheres in a neighborhood of to balls in a larger neighborhood.)
Conjecture 1.2 (see [Bw07], Question 1.6).
If is an irreducible lattice in a semisimple group of rank , then there is a nonempty -invariant subset such that and is undistorted up to dimension .
Here, represents the Hausdorff distance between the two sets.
Theorem 1.1 is a special case of this conjecture. As Bestvina, Eskin, and Wortman note in [BEW], Conj. 1.2 would imply that the th-order Dehn function of is bounded by . In recent years, a significant amount of progess has been made toward these conjectures. Druţu proved that a lattice of -rank 1 in a symmetric space of -rank has a Dehn function bounded by for any [Dru04], Leuzinger and Pittet proved that, conversely, any irreducible lattice in a symmetric space of rank 2 which is not cocompact has an exponentially large Dehn function [LP96], and the author proved Thurston’s conjecture in the case that [You].
In this paper, we make a step toward proving Conj. 1.2 by showing that, under some conditions on and , undistortedness follows from a Lipschitz extension property. We say that is Lipschitz -connected if there is a such that for any and any -Lipschitz map , there is a -Lipschitz extension . If , we say is Lipschitz -connected in if, under the above conditions, there is a -Lipschitz extension .
Suppose that is a nonempty closed subset with metric given by the restriction of the metric of . Suppose that is a geodesic metric space such that the Assouad-Nagata dimension of is finite. Suppose that one of the following is true:
is Lipschitz -connected.
is Lipschitz -connected, and if are the connected components of , then the sets are Lipschitz -connected with uniformly bounded implicit constant.
Then is undistorted up to dimension .
In the applications in this paper, will be a CAT(0) space (either a symmetric space or a building), and will either be a horosphere of or the complement of a set of disjoint horoballs.
When is CAT(0), a theorem of Gromov [Gro83, Wen08] implies that the th-order Dehn function of grows at most as fast as (i.e., if is a Lipschitz -cycle in , there is a Lipschitz -chain in such that and
If is CAT(0) and the hypotheses above hold, the th-order Dehn function of grows at most as fast as for .
This bound is often sharp; for instance, if there is a rank- flat of contained in , then the th-order Dehn function of grows at least as fast as .
We will apply Theorem 1.3 to find fillings in a family of solvable groups and in the Hilbert modular groups:
The group is Lipschitz -connected, is undistorted in up to dimension , and its th-order Dehn function is asymptotic to for .
This is a higher-dimensional version of a theorem of Gromov [Gro93, 5.A] which states that has quadratic Dehn function when . These bounds are sharp; there are -spheres in with volume but filling volume exponential in , so the th order Dehn function of is exponential [Gro93]. The same bounds apply to Hilbert modular groups:
If is a Hilbert modular group, then the th-order Dehn function of is asymptotic to for .
Let be a thick euclidean building and be an apartment. Then the vertices of form a lattice, and if is a geodesic ray, we say that has rational slope if it is parallel to a line segment connecting two vertices of . This condition is independent of the choice of , so if is a geodesic ray, we say it has rational slope if it has rational slope considered as a ray in some apartment . The boundary at infinity of consists of equivalence classes of geodesic rays, so if is a point in the boundary at infinity of , we say it has rational slope if one of the rays asymptotic to has rational slope. In particular, if the isometry group of acts cocompactly on a horosphere centered at , then has rational slope.
Let be a thick euclidean building and let be a point in its boundary at infinity which has rational slope and is not contained in a factor of rank less than (in particular, has rank at least ). Let be a horosphere in centered at . Then is Lipschitz -connected, undistorted in up to dimension , and its th-order Dehn function grows at most as fast as for .
is not -connected, so the bound on is sharp. Indeed, for every , there is a map such that is not null-homotopic in the -neighborhood of (see Lemma 4.15).
Note that if does not have rational slope, then may be -connected and locally Lipschitz -connected but not Lipschitz -connected. Cells of may intersect in arbitrarily small sets, and this can lead to arbitrarily small spheres which have small fillings in but filling volume in .
Theorem 1.7 is similar to Theorem 7.7 of [BW11], and gives a higher-order version of Theorem 1.1 of [Dru04] for buildings and products of buildings. (Though note that Theorem 1.1 of [Dru04] applies to -buildings as well as discrete buildings.)
The same methods lead to bounds on the higher-order Dehn functions of -arithmetic groups of -rank 1.
Let be a global function field, be a noncommutative, absolutely almost simple -group of -rank 1, let be a finite set of pairwise inequivalent valuations on , and let be the associated euclidean building. Then the th-order Dehn function of the -arithmetic group grows at most as fast as for .
This improves results of Bux and Wortman, who showed that is of type but not of type [BW11, BW07]. Bux and Wortman showed that horospheres in are -connected; Theorem 1.8 gives a quantitative proof of this fact.
Some possible other applications of Theorem 1.3 include the study of higher-order fillings in, for instance, metabelian groups, as in [dCT10], lattices of -rank 1 in semisimple groups, as in [Dru04], and -arithmetic lattices when is large, as in [BEW].
Notational conventions: If and are expressions, we will write if there is some constant such that . We write if there is some constant such that . When we wish to emphasize that depends on and , we write or . We give the round metric, scaled so that , and we define the standard -simplex to be the equilateral euclidean -simplex, scaled to have diameter 1.
Acknowledgements: This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada and by the Connaught Fund, University of Toronto. The author would like to thank MSRI and the organizers of the 2011 Quantitative Geometry program for their hospitality, and would like to thank Cornelia Druţu, Enrico Leuzinger, Romain Tessera, and Kevin Wortman for helpful discussions and suggestions.
2. Building fillings from simplices
The proof of Theorem 1.3 is based on the proof of a theorem of Lang and Schlichenmaier. Lang and Schlichenmaier proved:
Suppose that is a nonempty closed set and that . If is Lipschitz -connected, then there is a such that any Lipschitz map extends to a map with .
Here, is the Assouad-Nagata dimension of . The Assouad-Nagata dimension of is the smallest integer such that there is a such that for all , there is a covering of by sets of diameter at most (a -bounded covering) such that any set with diameter intersects at most sets in the cover (i.e., has -multiplicity at most ).
One consequence of Theorem 2.1 is that if is Lipschitz -connected for , then the identity map can be extended to a Lipschitz map and is a Lipschitz retract of . Consequently, if is a -cycle in and is a chain in with boundary , then is a chain in with boundary , so
and is undistorted in up to dimension . Theorem 1.3 claims that the same is true under the weaker condition that has finite Assouad-Nagata dimension.
Before we sketch the proof of Theorem 1.3, we need the notion of a quasi-conformal complex. We define a riemannian simplicial complex to be a simplicial complex with a metric which gives each simplex the structure of a riemannian manifold with corners. We say that such a complex is quasi-conformal (or that the complex is a QC complex) if there is a such that the riemannian metric on each simplex is -bilipschitz equivalent to a scaling of the standard simplex.
QC complexes are a compromise between the rigidity of simplicial complexes and the freedom of riemannian simplicial complexes. A key feature of simplicial complexes is that curves and cycles can be approximated by simplicial curves and cycles. This is not true in riemannian simplicial complexes, but it holds in QC complexes.
Specifically, a version of the Federer-Fleming deformation theorem holds in QC complexes. Recall that the Federer-Fleming theorem for simplicial complexes states that any Lipschitz cycle in a simplicial complex can be approximated by a simplicial cycle whose mass is comparable to the mass of . We will use the following variation of the Federer-Fleming theorem:
Let be a finite-dimensional scaled simplicial complex, that is, a simplicial complex where each simplex is given the metric of the standard simplex of diameter . There is a constant depending on such that if is a Lipschitz -cycle, then there are and such that
A proof of this theorem when can be found in [ECH92]. A simple scaling argument proves the general case. Note that, while the bound on depends on the size of the simplices, the bound on does not.
Because the bound on is independent of the size of the simplices in the complex, the following version of Theorem 2.2 holds for a QC complex:
Let be a QC complex. There is a constant depending on such that if is a Lipschitz -cycle, then there is a such that and .
Now we will sketch a proof of Theorem 1.3. Note that this sketch is incorrect due to some technical issues; we will fix these issues in the actual proof. In the proof of Theorem 1.5 of [LS05], Lang and Schlichenmaier show that, if , there are , and a cover of by subsets of such that:
every set with meets at most members of .
They then define functions ,
where , and show that these have the property that for any , there are no more than values of for which . Using these , they construct a Lipschitz map , where is the nerve of the supports of the . One can give the structure of a QC complex so that if is a simplex of with a vertex corresponding to , then . Since the diameter of is proportional to , this means that the parts of which are close to are given a fine triangulation and the parts of which are far from are given a coarse triangulation.
Since is Lipschitz -connected, one can construct a Lipschitz map , where is the -skeleton of . Then, if is an -cycle in , it has a filling in . We can use the Federer-Fleming theorem to approximate by some simplicial -chain which lies in . The pushforward of under will then be a filling of .
The problem with this argument is twofold. First, since is only defined on , we can’t define without extending to . We could define an appropriate metric on the disjoint union and a map , but this is no longer a simplicial complex. Second, since the cells of get arbitrarily small close to , may be an infinite sum of cells of .
We know of two ways to fix this issue. First, one can make sense of infinite sums of cells of by introducing Lipschitz currents [AK00]. The set of Lipschitz currents is a completion of the set of Lipschitz chains, and the defined above is a current in . Its pushforward is then a filling of . Second, we can change the construction of to avoid the problem. We take this approach in the rest of this section.
All the constants and all the implicit constants in and in this section will depend on , and .
First, we construct a QC complex which approximates . This complex will have geometry similar to on and it will have -small simplices on . For , let be the -neighborhood of .
There are such that if and , there is a covering of by sets , and functions ,
such that for any ,
if and , then is contained in a connected component of ,
the cover of by the sets has multiplicity at most , and
if , then
Let , , and be as in the Lang-Schlichenmaier construction above. Let We may assume that each is contained in a connected component of . Let , and let be the set
Then . Since , we can let be a -bounded covering of with -multiplicity at most , where is the constant in the definition of . Let and let .
Conditions (1) and (2) are easy to check. For (3), note that if , then , so lies in a single connected component of , and lies in the same component. For (4), note that if , then , so the cover has multiplicity at most . Likewise, if for some , then , where is the closed ball of radius around . Since has bounded -multiplicity, this can be true for only values of .
To check (5), suppose that . If , then . Otherwise, . But and , so , and . By symmetry, . ∎
Let be the nerve of the cover , with vertex set and let is the function such that and is linear on each simplex of . Define a riemannian metric on each simplex of by letting . If is a simplex of , then varies between and on , so this metric makes a QC complex.
There is a Lipschitz map with Lipschitz constant independent of . Furthermore, if for some , then is in the star of .
Consider the infinite simplex
with vertex set , so that is a subcomplex of . Let
where . The image of then lies in , and we can consider as a function .
It remains to show that is Lipschitz with respect to the QC metric on . Since is geodesic, it suffices to show that if and then . Let and be the minimal simplices of which contain and respectively. First, we claim that and share some vertex .
Let as above. If , then there is some such that and . Since is 1-Lipschitz, , so we can let . Otherwise, if , then there is some such that . We have
so , and as desired. We let .
Since and share , the value of on is at most , and
Furthermore, if , then
so has Lipschitz constant at most
Next, we construct a map on the -skeleton of . If is a simplex of , denote its vertex set by .
For any , there is a Lipschitz map with Lipschitz constant independent of which satisfies:
for every ,
if are the connected components of and
then for any simplex , we either have (if has a vertex of the form for some ) or for some (otherwise).
For each vertex , choose a point such that , and let . If , then , so and property (1) holds. We claim that this map is Lipschitz. Suppose that are vertices of . Then there is a path between them of length , and the Federer-Fleming theorem implies that this can be approximated by a path in the 1-skeleton of with . So, to check that is Lipschitz, it suffices to show that if and are connected by an edge , then .
We may assume that , so . Then we can bound by
Each term on the right is . For each term except , this follows from the remarks after the definition of . To bound , note that since there is an edge from to , there is a . Then and , so . Therefore, is Lipschitz.
It remains to check property (2). Let be a simplex of and suppose that for some . Then , so , and therefore, .
Otherwise, for all . Then there is some such that for all , and . ∎
If and are such that whenever and is a map with , there is an extension with , we say that is -locally Lipschitz -connected.
If and satisfy the conditions of Theorem 1.3 and is sufficiently small, then there is a Lipschitz extension with Lipschitz constant independent of such that for every .
In this proof, it will be convenient to let be the boundary of the standard -simplex and be the standard -simplex. If , we let and be scalings of and . If a space is Lipschitz -connected, there is a such that if , any Lipschitz map can be extended to a Lipschitz map with . By scaling, any Lipschitz map can be extended to a Lipschitz map with
If is Lipschitz -connected, then we can use Lipschitz -connectivity to extend . That is, if we have already defined on and is a -simplex, then the Riemannian metric on is bilipschitz equivalent to for any . Since is a Lipschitz map of a -sphere, we can extend over , and the extension satisfies .
If is not Lipschitz -connected, we need a more careful approach. By hypothesis, is Lipschitz -connected; let be the constant in the definition of Lipschitz -connectivity.
Let and let . If is a map with , we claim that can be extended to a Lipschitz map on . If for some , then we can extend to using the Lipschitz -connectivity of . Otherwise, there is some such that . Since , the image of is contained in . Therefore, is -locally Lipschitz -connected.
If is a simplex, we say that it is coarse if all its vertices are of the form for . We say that it is fine if it has a vertex of the form for some ; all fine simplices have diameter and all coarse ones have diameter . By the previous lemma, we can choose so that for every coarse simplex , there is some such that . If is the subcomplex consisting of coarse simplices, we can extend to a map by induction; if is defined, then for some . We extend over using the Lipschitz -connectivity of . The Lipschitz constant of is bounded independently of .
Again by the previous lemma, we may choose sufficiently small that any fine simplex has diameter . We can then extend over the fine simplices of using the local Lipschitz connectivity of to get the desired map .
In either case, if , then only if . In particular, is contained in a fine simplex of diameter and , so
as desired. ∎
Therefore, has small displacement. To complete the proof of Theorem 1.3, we will need a lemma concerning such maps:
Suppose that , that is a Lipschitz -cycle in , that is -locally Lipschitz -connected, and that . For any , there is a such that if is a -Lipschitz map with displacement (i.e., for all ), then
Since is locally Lipschitz -connected, if is a simplicial -complex, is a subcomplex, and is a map with sufficiently small Lipschitz constant, then there is an extension with Lipschitz constant . Write as a sum of Lipschitz maps . Let be the maximum Lipschitz constant of the ’s. In the following calculations, all our implicit constants will depend on , , , , and . We claim that
First, we can subdivide into simplices each with diameter . We can use this subdivision to replace with a sum where and each has Lipschitz constant at most .
There is a simplicial -complex with at most top-dimensional faces, a simplicial cycle on , and a map with such that the restriction of to each top-dimensional face of is one of the ’s and . Define by letting and . Then , and if is sufficiently small, we can extend it to a Lipschitz map with . This is a homotopy from to , so the push-forward of is a filling of with mass
as desired. ∎
Proof of Theorem 1.3.
Suppose that is a -cycle in and is a -chain filling it. Let be the subcomplex of spanned by the vertices . Then , and is a cycle in with mass . Each simplex of has diameter , so by Thm. 2.2, there is a depending only on , a simplicial cycle approximating , and a chain such that and .
Then is a -chain in with boundary and mass
Thm. 2.3 lets us approximate this by a chain
with boundary .
By Lemma 2.8, if , then for sufficiently small, there is a Lipschitz -chain in such that
and . Let
as desired. ∎
The rest of this paper is dedicated to applying this theorem to horospheres and lattices in symmetric spaces and buildings.
3. Fillings in
Theorem 1.3 is useful because it reduces a difficult-to-prove statement about the undistortedness of an inclusion to an easier-to-prove Lipschitz extension property. For example, in this section, we will prove:
The solvable Lie group is Lipschitz -connected.
We start by defining , . This group is a solvable Lie group which can be written as a semidirect product of and , where acts on as the group of diagonal matrices with positive coefficients and determinant 1. When , this is the three-dimensional solvable group corresponding to solvegeometry.
All the constants and implicit constants in this section will depend on .
One feature of this group is that it can be realized as a horosphere in a product of hyperbolic planes. Let be the hyperbolic plane and let be a Busemann function for . We can define Busemann functions in the product by letting . Then is a Busemann function for , and acts freely, isometrically, and transitively on the resulting horosphere . The metric induced on by inclusion in is bilipschitz equivalent to a left-invariant Riemannian metric on .
This group also appears as a subgroup of a Hilbert modular group. If is a Hilbert modular group and , then there is a collection of disjoint open horoballs in such that the boundary of each horosphere is bilipschitz equivalent to and acts cocompactly on [Pit95]. Consequently, Theorem 1.6 is also a corollary of Theorem 3.1.
Let be a metric space, let be the infinite-dimensional simplex with vertex set , and let be its -skeleton. Let denote the -simplex with vertices . Then is Lipschitz -connected if and only if there exists a map such that
For all , .
There is a such that for any and any simplex , we have
One direction is clear; if is Lipschitz -connected, then one can construct by letting for all , then using the Lipschitz connectivity of to extend over each skeleton inductively.
The other direction is an application of the Whitney decomposition. We view as a subset of ; by the Whitney covering lemma, the interior of can be decomposed into a union of countably many dyadic cubes such that for each cube , one has . We can decompose each cube into boundedly many simplices to get a triangulation of the interior of where each simplex is bilipschitz equivalent to a scaling of the standard simplex.
We construct a map using this triangulation. For each vertex in , let be a point in such that is minimized. One can check that is a Lipschitz map from , so