Lipschitz connectivity and filling invariants in solvable groups and buildings

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 $k$th-order Dehn function bounded by $V^{(k+1)/k}$ 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 $k$-sphere of volume $V$ 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 $k$.

One way around this is to consider Lipschitz extension properties. A typical Lipschitz extension property is Lipschitz $k$-connectivity; we say that a space $X$ is Lipschitz $k$-connected (with constant $c$) if there is a $c$ such that for any $0\leq d\leq k$ and any $l$-Lipschitz map $f:S^{d}\to X$, there is a $cl$-Lipschitz extension $\bar{f}:D^{d+1}\to X$. 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 $k$-connected for any $k$. Any map $f:S^{d}\to X$ can be extended to a map $\bar{f}:D^{d+1}\to X$ by coning $f$ off to a point along geodesics, and if $f$ is Lipschitz, so is $\bar{f}$.

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 $n$-connected subsets of spaces with finite Assouad-Nagata dimension. Recall that Lubotzky, Mozes, and Raghunathan proved that

One way to state this theorem is that the inclusion $\Gamma\hookrightarrow G$ does not induce any distortion of lengths. That is, there is a $c>0$ such that if $x,y\in\Gamma$ are connected by a path of length $l$ in $G$, then they are connected by a path of length $\leq cl$ in the Cayley graph of $\Gamma$. We can think of this as an efficient 1-dimensional filling of a 0-sphere. Many authors have conjectured that when $G$ has higher rank, we can fill higher-dimensional spheres efficiently; for example, Thurston famously conjectured that $\operatorname{SL}(n;\mathbb{Z})$ has quadratic Dehn function for $n\geq 4$ [Ger93], and Gromov conjectured that the $(k-2)$-th order Dehn function of a lattice in a symmetric space of rank $k$ 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 $d$-chain in $Y$ is a formal sum of Lipschitz maps $\Delta_{d}\to Y$. One can define the boundary of a Lipschitz chain as for singular chains, and this gives rise to a homology theory. If $\alpha$ is a Lipschitz $d$-cycle in $Y$, define

to be the filling volume of $\alpha$ in $Y$. In particular, if $Y$ is a geodesic metric space and $\alpha$ is the 0-cycle $x-y$, then $\operatorname{FV}^{1}_{Y}(\alpha)=d(x,y)$.

If $Z\subset X$, we say that $Z$ is undistorted up to dimension $n$ if there is some $r\geq 0$ and $c$ such that if $\alpha$ is a Lipschitz $d$-cycle in $Z$ and $d<n$, 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 $Z$ to balls in a larger neighborhood.)

Here, $d_{\text{Haus}}(Z,\Gamma)$ 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 $k$th-order Dehn function of $\Gamma$ is bounded by $V^{(k+1)/k}$. In recent years, a significant amount of progess has been made toward these conjectures. Druţu proved that a lattice of $\mathbb{Q}$-rank 1 in a symmetric space of $\mathbb{R}$-rank $\geq 3$ has a Dehn function bounded by $n^{2+\epsilon}$ for any $\epsilon>0$ [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 $n\geq 5$ [You].

In this paper, we make a step toward proving Conj. 1.2 by showing that, under some conditions on $G$ and $\Gamma$, undistortedness follows from a Lipschitz extension property. We say that $Z$ is Lipschitz $n$-connected if there is a $c$ such that for any $0\leq d\leq k$ and any $l$-Lipschitz map $f:S^{d}\to Z$, there is a $cl$-Lipschitz extension $\bar{f}:D^{d+1}\to Z$. If $Z\subset Y$, we say $Z$ is Lipschitz $n$-connected in $Y$ if, under the above conditions, there is a $cl$-Lipschitz extension $\bar{f}:D^{d+1}\to Y$.

In the applications in this paper, $X$ will be a CAT(0) space (either a symmetric space or a building), and $Z$ will either be a horosphere of $X$ or the complement of a set of disjoint horoballs.

When $X$ is CAT(0), a theorem of Gromov [Gro83, Wen08] implies that the $k$th-order Dehn function of $X$ grows at most as fast as $V^{(k+1)/k}$ (i.e., if $\alpha$ is a Lipschitz $k$-cycle in $X$, there is a Lipschitz $(k+1)$-chain $\beta$ in $X$ such that $\partial\beta=\alpha$ and

Therefore,

This bound is often sharp; for instance, if there is a rank-$(k+1)$ flat of $X$ contained in $Z$, then the $k$th-order Dehn function of $Z$ grows at least as fast as $V^{(k+1)/k}$.

We will apply Theorem 1.3 to find fillings in a family of solvable groups and in the Hilbert modular groups:

This is a higher-dimensional version of a theorem of Gromov [Gro93, 5.A₉] which states that $\operatorname{Sol}_{2n-1}$ has quadratic Dehn function when $n>1$. These bounds are sharp; there are $n$-spheres in $\operatorname{Sol}_{2n-1}$ with volume $V$ but filling volume exponential in $V$, so the $n$th order Dehn function of $\operatorname{Sol}_{2n-1}$ is exponential [Gro93]. The same bounds apply to Hilbert modular groups:

We will also apply the methods of Theorem 1.3 to horospheres in euclidean buildings and to the $S$-arithmetic groups considered in [BW11].

Let $X$ be a thick euclidean building and $E\subset X$ be an apartment. Then the vertices of $E$ form a lattice, and if $r:[0,\infty)\to E$ is a geodesic ray, we say that $r$ has rational slope if it is parallel to a line segment connecting two vertices of $E$. This condition is independent of the choice of $E$, so if $r:[0,\infty)\to X$ is a geodesic ray, we say it has rational slope if it has rational slope considered as a ray in some apartment $E$. The boundary at infinity of $X$ consists of equivalence classes of geodesic rays, so if $\tau\in X_{\infty}$ is a point in the boundary at infinity of $X$, we say it has rational slope if one of the rays asymptotic to $\tau$ has rational slope. In particular, if the isometry group of $X$ acts cocompactly on a horosphere centered at $\tau$, then $\tau$ has rational slope.

$Z$ is not $(n-1)$-connected, so the bound on $k$ is sharp. Indeed, for every $r>0$, there is a map $\alpha:S^{n-1}\to Z$ such that $\alpha$ is not null-homotopic in the $r$-neighborhood of $Z$ (see Lemma 4.15).

Note that if $\tau$ does not have rational slope, then $Z$ may be $(n-2)$-connected and locally Lipschitz $(n-2)$-connected but not Lipschitz $(n-2)$-connected. Cells of $X$ may intersect $Z$ in arbitrarily small sets, and this can lead to arbitrarily small spheres which have small fillings in $X$ but filling volume $\sim 1$ in $Z$.

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 $\mathbb{R}$-buildings as well as discrete buildings.)

The same methods lead to bounds on the higher-order Dehn functions of $S$-arithmetic groups of $K$-rank 1.

This improves results of Bux and Wortman, who showed that $G(\mathcal{O}_{S})$ is of type $F_{\dim X-1}$ but not of type $F_{\dim X}$ [BW11, BW07]. Bux and Wortman showed that horospheres in $X$ are $(\dim X-2)$-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 $\mathbb{Q}$-rank 1 in semisimple groups, as in [Dru04], and $S$-arithmetic lattices when $|S|$ is large, as in [BEW].

The proof of Theorem 1.3 is based on the proof of a theorem of Lang and Schlichenmaier. Lang and Schlichenmaier proved:

Here, $\dim_{\text{AN}}(X)$ is the Assouad-Nagata dimension of $X$. The Assouad-Nagata dimension of $X$ is the smallest integer such that there is a $c$ such that for all $s>0$, there is a covering $\mathcal{B}_{s}$ of $X$ by sets of diameter at most $cs$ (a $cs$-bounded covering) such that any set with diameter $\leq s$ intersects at most $n+1$ sets in the cover (i.e., $\mathcal{B}_{s}$ has $s$-multiplicity at most $n+1$).

One consequence of Theorem 2.1 is that if $Z$ is Lipschitz $n$-connected for $n=\dim_{\text{AN}}(X)$, then the identity map $Z\to Z$ can be extended to a Lipschitz map $\bar{f}:X\to Z$ and $Z$ is a Lipschitz retract of $X$. Consequently, if $\alpha$ is a $(k-1)$-cycle in $Z$ and $\beta$ is a chain in $X$ with boundary $\alpha$, then $\bar{f}_{\sharp}(\beta)$ is a chain in $Z$ with boundary $\alpha$, so

and $Z$ is undistorted in $X$ up to dimension $n$. Theorem 1.3 claims that the same is true under the weaker condition that $X$ 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 $c$ such that the riemannian metric on each simplex is $c$-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 $a$ in a simplicial complex can be approximated by a simplicial cycle $P(a)$ whose mass is comparable to the mass of $a$. We will use the following variation of the Federer-Fleming theorem:

A proof of this theorem when $s=1$ can be found in [ECH⁺92]. A simple scaling argument proves the general case. Note that, while the bound on $\operatorname{mass}Q(a)$ depends on the size of the simplices, the bound on $\operatorname{mass}P(a)$ does not.

Because the bound on $\operatorname{mass}P(a)$ is independent of the size of the simplices in the complex, the following version of Theorem 2.2 holds for a QC complex:

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 $\dim_{\text{AN}}(X)<\infty$, there are $a>0$, $0<b<1$ and a cover $\mathcal{B}=(B_{i})_{i\in I_{0}}$ of $X\setminus Z$ by subsets of $X\setminus Z$ such that:

1. (1)

   $\operatorname{diam}B_{i}\leq ad(B_{i},Z)$ for every $i\in I_{0}$

2. (2)

   every set $D\subset X\setminus Z$ with $\operatorname{diam}D\leq bd(D,Z)$ meets at most $\dim_{\text{AN}}(X)+1$ members of $(B_{i})_{i\in I_{0}}$.

They then define functions $\sigma_{i}:X\setminus Z\to\mathbb{R}$,

where $\delta=b/(2(b+1))$, and show that these have the property that for any $x$, there are no more than $\dim_{\text{AN}}(X)+1$ values of $i$ for which $\sigma_{i}(x)>0$. Using these $\sigma_{i}$, they construct a Lipschitz map $g:X\setminus Z\to\Sigma_{0}$, where $\Sigma_{0}$ is the nerve of the supports of the $\sigma_{i}$. One can give $\Sigma_{0}$ the structure of a QC complex so that if $\Delta$ is a simplex of $\Sigma_{0}$ with a vertex corresponding to $\sigma_{i}$, then $\operatorname{diam}\Delta\sim\operatorname{diam}\operatorname{supp}\sigma_{i}$. Since the diameter of $\operatorname{supp}\sigma_{i}$ is proportional to $d(\sigma_{i},Z)$, this means that the parts of $\Sigma_{0}$ which are close to $Z$ are given a fine triangulation and the parts of $\Sigma_{0}$ which are far from $Z$ are given a coarse triangulation.

Since $Z$ is Lipschitz $n$-connected, one can construct a Lipschitz map $h:\Sigma_{0}^{(n+1)}\to Z$, where $\Sigma_{0}^{(n+1)}$ is the $(n+1)$-skeleton of $\Sigma_{0}$. Then, if $\alpha$ is an $n$-cycle in $Z$, it has a filling $\beta$ in $X$. We can use the Federer-Fleming theorem to approximate $g_{\sharp}(\beta)$ by some simplicial $(n+1)$-chain $P(\beta)$ which lies in $\Sigma_{0}^{(n+1)}$. The pushforward of $P(\beta)$ under $h$ will then be a filling of $\alpha$.

The problem with this argument is twofold. First, since $g$ is only defined on $X\setminus Z$, we can’t define $g_{\sharp}(\beta)$ without extending $g$ to $Z$. We could define an appropriate metric on the disjoint union $\Sigma_{0}\amalg Z$ and a map $X\to\Sigma_{0}\amalg Z$, but this is no longer a simplicial complex. Second, since the cells of $\Sigma$ get arbitrarily small close to $Z$, $P(\beta)$ may be an infinite sum of cells of $\Sigma$.

We know of two ways to fix this issue. First, one can make sense of infinite sums of cells of $\Sigma$ by introducing Lipschitz currents [AK00]. The set of Lipschitz currents is a completion of the set of Lipschitz chains, and the $P(\beta)$ defined above is a current in $\Sigma_{0}\amalg Z$. Its pushforward is then a filling of $\alpha$. Second, we can change the construction of $\Sigma_{0}$ to avoid the problem. We take this approach in the rest of this section.

All the constants and all the implicit constants in $\lesssim$ and $\sim$ in this section will depend on $X,Z$, and $n$.

First, we construct a QC complex $\Sigma$ which approximates $X$. This complex will have geometry similar to $\Sigma_{0}$ on $X\setminus Z$ and it will have $\epsilon$-small simplices on $Z$. For $t>0$, let $N_{t}(Z)\subset X$ be the $t$-neighborhood of $Z$.

Let $\Sigma$ be the nerve of the cover $\{\operatorname{supp}\tau_{k}\}_{k\in K}$, with vertex set $\{v_{k}\}_{k\in K}$ and let $s:\Sigma\to\mathbb{R}$ is the function such that $s(v_{k})=r(k)$ and $s$ is linear on each simplex of $\Sigma$. Define a riemannian metric $x_{c}$ on each simplex of $\Sigma$ by letting $dx_{c}^{2}=s^{2}\;dx^{2}$. If $S=\langle v_{k_{1}},\dots v_{k_{n}}\rangle$ is a simplex of $\Sigma$, then $s$ varies between $\gamma^{-1}r(k_{1})$ and $\gamma r(k_{1})$ on $S$, so this metric makes $\Sigma$ a QC complex.

Next, we construct a map $h:\Sigma^{(n+1)}\to Z$ on the $(n+1)$-skeleton of $\Sigma$. If $\Delta$ is a simplex of $\Sigma$, denote its vertex set by $\mathcal{V}(\Delta)$.

If $\epsilon>0$ and $n$ are such that whenever $k\leq n$ and $\tau:S^{k}\to Z$ is a map with $\operatorname{Lip}\tau\leq\epsilon$, there is an extension $\bar{\tau}:D^{k+1}\to Z$ with $\operatorname{Lip}\bar{\tau}\lesssim\operatorname{Lip}\tau$, we say that $Y$ is $\epsilon$-locally Lipschitz $n$-connected.