# Characterizations of higher rank hyperbolicity

Tommaso Goldhirsch  Department of Mathematics
ETH Zürich
Rämistrasse 101
8092 Zürich
Switzerland<EMAIL_ADDRESS>and  Urs Lang  Department of
Mathematics
ETH Zürich
Rämistrasse 101
8092 Zürich
Switzerland<EMAIL_ADDRESS>
(Date: December 30, 2021)

Research supported by Swiss National Science Foundation Grant 197090.

##  1\. Introduction

The concept of Gromov hyperbolicity manifests itself in many different ways.
With only mild assumptions on the underlying metric space, the spectrum of
equivalent properties includes various thin triangle conditions, the stability
of quasi-geodesics (the Morse lemma), a linear isoperimetric inequality for
$1$-cycles, and a sub-quadratic isoperimetric inequality [2, 3, 4, 5, 6, 8,
14, 18, 28, 29, 30]. We present a similar list of six equivalent properties in
the context of generalized non-positive curvature and higher asymptotic rank.
This complements the results in [34] and in the recent paper [24]. We give a
largely self-contained proof, providing some improvements and simplifications
for the known part.

The setup is as in [24]. We assume, for simplicity, that the metric space
$X=(X,d)$ is proper. For $n\geq 1$, we say that $X$ satisfies condition $({\rm
CI}_{n})$ if there is a constant $c$ such that any two points $x,y$ in $X$ can
be joined by a curve of length $\leq c\,d(x,y)$, and for $k=1,\ldots,n$, every
$k$-cycle $R$ in some $r$-ball is the boundary of a $(k+1)$-chain $S$ with
mass

${\mathbf{M}}(S)\leq c\,r\,{\mathbf{M}}(R).$

We use the chain complex ${\mathbf{I}}_{\ast,\text{\rm c}}(X)$ of metric
integral currents with compact support (see Sect. 2). The cone inequalities
$({\rm CI}_{n})$ hold in particular if $X$ is a $\operatorname{CAT}(0)$ space
or a space with a conical geodesic bicombing [10, 31]. Every hyperbolic group
acts geometrically on a proper polyhedral complex with such a structure [26]
(see also [7, 21] for more groups with this property).

The asymptotic rank of $X$ is the supremum of all $k\geq 0$ for which there
exist a sequence $0<r_{i}\to\infty$ and subsets $Y_{i}\subset X$ such that the
rescaled sets $(Y_{i},\frac{1}{r_{i}}d)$ converge in the Gromov–Hausdorff
topology to the unit ball in some $k$-dimensional normed space. This is a
quasi-isometry invariant, and if $X$ is a geodesic metric space satisfying
$({\rm CI}_{1})$, then the asymptotic rank is at most $1$ if and only if $X$
is Gromov hyperbolic [34]. If $X$ is a cocompact $\operatorname{CAT}(0)$ space
or a cocompact space with a conical geodesic bicombing, then the asymptotic
rank agrees with the maximal dimension of an isometrically embedded Eulidean
or normed space, respectively [9, 23].

As higher-dimensional analogs of quasi-geodesics we use quasi-minimizing
$n$-chains satisfying a uniform density bound at large scales. Let
$S\in{\mathbf{I}}_{n,\text{\rm c}}(X)$. For constants $C\geq 1$ and $a\geq 0$,
we say that $S$ has $(C,a)$-controlled density if for all $x\in X$ and $r>a$,
the piece of $S$ in the closed $r$-ball at $x$ has mass at most $Cr^{n}$, or

$\Theta_{x,r}(S):=\frac{1}{r^{n}}{\mathbf{M}}(S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(r))\leq
C.$

For almost every $r>0$, the boundary
$\partial(S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(r))$
has finite mass and
$S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(r)$
is itself an element of ${\mathbf{I}}_{n,\text{\rm c}}(X)$ (see again Sect.
2).

Suppose that $Y\subset X$ is a closed set containing the support
$\operatorname{spt}(\partial S)$ of $\partial S$. For $Q\geq 1$ and $a\geq 0$,
we say that $S$ is $(Q,a)$-quasi-minimizing mod $Y$ if for every point
$x\in\operatorname{spt}(S)$ at a distance $b>a$ from $Y$, the inequality

${\mathbf{M}}(S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(r))\leq
Q\,{\mathbf{M}}(T)$

holds for almost all $r\in(a,b)$ and all $T\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$ with $\partial
T=\partial(S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(r))$.
A current $S\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ is called a $(Q,a)$-quasi-
minimizer if $S$ is $(Q,a)$-quasi-minimizing mod $\operatorname{spt}(\partial
S)$. As for the analogy with quasi-geodesics, one can easily check that for a
Lipschitz quasi-isometric embedding $\gamma\colon[0,t]\to X$ of an interval,
the associated current $\gamma_{\\#}\llbracket
0,t\rrbracket\in{\mathbf{I}}_{1,\text{\rm c}}(X)$ is a quasi-minimizer with
controlled density (compare the case $n=1$ of Proposition 6.1). For $n>1$,
quasi-minimizers offer more flexibility than quasiflats.

We can now state our main result.

######  Theorem 1.1.

Suppose that $X$ is a proper metric satisfying condition (CIn) for some $n\geq
1$. Then the following six properties are equivalent:

  1. $({\rm AR}_{n})$

(asymptotic rank) the asymptotic rank of $X$ is at most $n$;

  2. $({\rm SII}_{n})$

(sub-Euclidean isoperimetric inequality) for all $\epsilon>0$ there is a
constant $M_{0}>0$ such that every cycle $Z\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$ is the boundary of a $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ with
mass ${\mathbf{M}}(V)<\epsilon\max\\{M_{0},{\mathbf{M}}(Z)\\}^{(n+1)/n}$;

  3. $({\rm LII}_{n})$

(linear isoperimetric inequality) there is a constant $\nu>0$, and for all
$C>0$ there is a $\lambda>0$, such that every cycle
$Z\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ with $(C,a)$-controlled density bounds
a $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ with
${\mathbf{M}}(V)\leq\max\\{\lambda,\nu a\\}\,{\mathbf{M}}(Z)$;

  4. $({\rm ML}_{n})$

(Morse lemma) For all $C>0$ and $Q\geq 1$ there is a constant $l\geq 0$ such
that if $Z\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ is a cycle with
$(C,a)$-controlled density and $Y\subset X$ is a closed set such that $Z$ is
$(Q,a)$-quasi-minimizing mod $Y$, then $\operatorname{spt}(Z)$ is within
distance at most $\max\\{l,4a\\}$ from $Y$;

  5. $(\SS_{n})$

(slim simplices) for all $L\geq 1$ there is a constant $D\geq 0$ such that if
$\Delta$ is a Euclidean $(n+1)$-simplex and $f\colon\partial\Delta\to X$ is a
map whose restriction to each facet of $\Delta$ is an $(L,a)$-quasi-isometric
embedding, then the image of every facet is within distance at most $D(1+a)$
of the union of the images of the remaining ones;

  6. $({\rm FR}_{n})$

(filling radius) for all $C>0$ there is a constant $h>0$ such that every cycle
$Z\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ with $(C,a)$-controlled density bounds
a $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ whose support is within distance at
most $\max\\{h,a\\}$ from $\operatorname{spt}(Z)$.

Note that for $n=1$, $({\rm SII}_{n})$ corresponds to a sub-quadratic
inequality. The equivalence of $({\rm AR}_{n})$ and $({\rm SII}_{n})$ was
established in [34] in a more general setup for complete metric spaces. The
proof of the forward implication uses an elaborate thick-thin decomposition
for integral cycles from [33] and also the non-trivial fact that a weakly
convergent sequence of cycles converges with respect to the filling volume
[32]. We review the entire argument. Employing an elegant new variational
result from [20] and introducing a more quantitative approach for the
convergence of cycles, we reduce the overall complexity substantially. In
fact, we prove the sub-Euclidean isoperimetric inequality first in a somewhat
restricted form (Theorem 5.1, compare Theorem 4.4 in [24]) and then deduce
$({\rm LII}_{n})$ and $({\rm SII}_{n})$.

Assertion $({\rm LII}_{n})$ is new. In symmetric spaces of non-compact type
and rank $\leq n$, a linear isoperimetric inequality holds for all $n$-cycles
(see p. 105 in [19], and [27]), and it is a long-standing problem whether this
generalizes, for instance, to cocompact Hadamard manifolds or
$\operatorname{CAT}(0)$ spaces of (asymptotic) rank $\leq n$. In our
statement, the isoperimetric constant depends on the density bound, so Theorem
1.1 does not resolve this; nevertheless, $({\rm LII}_{n})$ turns out to be
equivalent to the remaining properties.

The proofs of Theorem 5.1 and Theorem 5.2 in [24] show that $({\rm
SII}_{n})\Rightarrow({\rm ML}_{n})\Rightarrow(\SS_{n})$, except for a less
explicit distance bound in the slim simplex property. The second step involves
a technical result approximating quasiflats by quasi-minimizers. We go through
the argument in detail, keeping track of the dependence of constants, and thus
showing that the bound is linear in the coarseness parameter $a$. This fact is
used in the proof of the backward implication $(\SS_{n})\Rightarrow({\rm
AR}_{n})$.

The statement of $({\rm ML}_{n})$ differs formally from the usual stability
assertion for quasi-geodesics, but is versatile. If
$S\in{\mathbf{I}}_{n,c}(X)$ with $\operatorname{spt}(\partial S)\subset Y$ is
quasi-minimizing mod $Y$, then the extra assumption we need in order to
conclude that $S$ is confined to a bounded neighborhood of $Y$ is that $S$ can
be closed up to a cycle $Z=S-S^{\prime}$ with controlled density and with
$\operatorname{spt}(S^{\prime})\subset Y$. Note that there is no
(quasi-)minimality assumption on $S^{\prime}$; the density bound suffices.
However, if $S_{1},S_{2}\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ are two
$(Q,a)$-quasi-minimizers with $\partial S_{1}=\partial S_{2}$, each with
$(C,a)$-controlled density, then $S_{1}$ is $(Q,a)$-quasi-minimizing mod
$\operatorname{spt}(S_{2})$ and vice-versa, so $({\rm ML}_{n})$ implies that
the Hausdorff distance between the supports is bounded by $\max\\{l,4a\\}$ for
$l=l(2C,Q)$.

The last assertion of Theorem 1.1 is yet another way of expressing that
$n$-cycles with controlled density (such as geodesic triangles if $n=1$) are
thin. We use an iterative application of the sub-Euclidean isoperimetric
inequality to show that the conclusion of $({\rm FR}_{n})$ holds for every
mass minimizing $V$ with $\partial V=Z$. In [34], the filling radius was used
to prove that $({\rm SII}_{n})\Rightarrow({\rm AR}_{n})$. Similarly, $({\rm
FR}_{n})\Rightarrow({\rm AR}_{n})$.

The paper is organized as follows. In Sect. 2 we recall the definition of
metric currents and collect some basic results. In Sect. 3 we first review an
approximation result for cycles from [20] and then use this to give a short
proof of a quantitative version of a result from [32], showing that cycles
with bounded mass and sufficiently small uniformly bounded density at some
fixed scale have small filling volume. We use this further in Sect. 4 to
discuss the convergence of cycles. Sect. 5 is then devoted to isoperimetric
inequalities and shows in particular that

$({\rm AR}_{n})\Rightarrow({\rm LII}_{n})\Leftrightarrow({\rm SII}_{n}).$

In Sect. 6 we prove more explicit versions of two propositions from [24]
relating quasiflats and quasi-minimizers. The concluding Sect. 7 then shows in
particular that

$({\rm SII}_{n})\Rightarrow({\rm ML}_{n})\Rightarrow(\SS_{n})\Rightarrow({\rm
AR}_{n})\quad\text{and}\quad({\rm SII}_{n})\Rightarrow({\rm
FR}_{n})\Rightarrow({\rm AR}_{n}).$

In fact, we prove all implications (and hence Theorem 1.1) in a stronger form,
for any class of proper metric spaces satisfying the respective assumptions
uniformly, and with constants depending only on the data involved and on the
class, rather than on individual members (see Sect. 5 and Sect. 7).

What is missing from the list in Theorem 1.1 is a rank $n$ analog of Gromov’s
quadruple definition of $\delta$-hyperbolicity ([18], p. 89). A $2(n+1)$-point
condition of this type will be investigated in [22].

### Acknowledgement

We thank Stefan Wenger for a useful discussion.

##  2\. Preliminaries

Currents with finite mass in metric spaces were introduced in [1]. Here, for
consistency with [24], on which we heavily rely, we will work with the local
theory described in [25]. However, for the class of integral currents with
compact support, the principal objects in this paper, the formal difference
between the two approaches is marginal.

### Currents

An integral $n$-current may roughly be thought of as an oriented
$n$-dimensional Lipschitz surface equipped with a summable integer density
function. Formally though, $n$-currents are defined as functionals; on
compactly supported differential $n$-forms in the classical case (going back
to de Rham), and on suitable $(n+1)$-tuples of real-valued (locally) Lipschitz
functions for non-smooth ambient spaces. The relating principle (proposed by
De Giorgi) is that the tuple $(f_{0},\ldots,f_{n})$, say if the $f_{i}$ are
smooth functions on ${\mathbb{R}}^{N}$, represents the form
$f_{0}\,df_{1}\wedge\ldots\wedge df_{n}$.

We assume that the underlying metric space $X=(X,d)$ is proper (hence complete
and separable). For $n\geq 0$, we let ${\mathscr{D}}^{n}(X)$ denote the set of
all $(n+1)$-tuples $(f_{0},\ldots,f_{n})$ of Lipschitz functions $f_{i}\colon
X\to{\mathbb{R}}$ such that $f_{0}$ has compact support
$\operatorname{spt}(f_{0})$ (in [25], $f_{1},\ldots,f_{n}$ are merely locally
Lipschitz, but the following definition is equivalent). An $n$-dimensional
current $S$ in $X$ is a function $S\colon{\mathscr{D}}^{n}(X)\to{\mathbb{R}}$
satisfying the following three conditions:

  1. (1)

$S$ is $(n+1)$-linear;

  2. (2)

$S(f_{0,k},\ldots,f_{n,k})\to S(f_{0},\ldots,f_{n})$ whenever $f_{i,k}\to
f_{i}$ pointwise on $X$ with uniformly bounded Lipschitz constants
($i=0,\dots,n$) and with $\bigcup_{k}\operatorname{spt}(f_{0,k})\subset K$ for
some compact set $K\subset X$;

  3. (3)

$S(f_{0},\ldots,f_{n})=0$ whenever one of the functions $f_{1},\ldots,f_{n}$
is constant on a neighborhood of $\operatorname{spt}(f_{0})$.

It follows from these axioms that $S$ is alternating in the last $n$
arguments. The vector space of all $n$-dimensional currents in $X$ is denoted
${\mathscr{D}}_{n}(X)$. Every function $w\in L^{1}_{\text{\rm
loc}}({\mathbb{R}}^{n})$ induces a current $\llbracket
w\rrbracket\in{\mathscr{D}}_{n}({\mathbb{R}}^{n})$ defined by

$\llbracket w\rrbracket(f_{0},\dots,f_{n}):=\int
wf_{0}\det\bigl{[}\partial_{j}f_{i}\bigr{]}_{i,j=1}^{n}\,dx;$

note that the partial derivatives $\partial_{j}\pi_{i}$ exist almost
everywhere by Rademacher’s theorem. For a Borel set $W\subset{\mathbb{R}}^{n}$
we put $\llbracket W\rrbracket:=\llbracket\chi_{W}\rrbracket$, where
$\chi_{W}$ denotes the characteristic function.

### Support, push-forward, and boundary

Let $S\in{\mathscr{D}}_{n}(X)$. There exists a smallest closed subset of $X$,
the support $\operatorname{spt}(S)$ of $S$, such that the value
$S(f_{0},\ldots,f_{n})$ depends only on the restrictions of
$f_{0},\dots,f_{n}$ to this set. Thus, for any closed set $D\subset X$
containing $\operatorname{spt}(S)$, $S$ induces a current in
${\mathscr{D}}_{n}(D)$, still denoted by $S$. For a proper Lipschitz map
$\phi\colon D\to Y$ into another proper metric space $Y$, the push-forward
$\phi_{\\#}S\in{\mathscr{D}}_{n}(Y)$ is the current with support in
$\phi(\operatorname{spt}(S))$ defined by

$(\phi_{\\#}S)(f_{0},\ldots,f_{n}):=S(f_{0}\circ\phi,\ldots,f_{n}\circ\phi)$

for all $(f_{0},\ldots,f_{n})\in{\mathscr{D}}^{n}(Y)$. In the simplest case,
if $\llbracket a,b\rrbracket:=\llbracket[a,b]\rrbracket$ is the current in
${\mathscr{D}}_{1}({\mathbb{R}})$ (or ${\mathscr{D}}_{1}([a,b])$) associated
with an interval, and if $\gamma\colon[a,b]\to X$ is a Lipschitz curve, then

$\gamma_{\\#}\llbracket a,b\rrbracket(f_{0},f_{1})=\llbracket
a,b\rrbracket(f_{0}\circ\gamma,f_{1}\circ\gamma)=\int_{a}^{b}(f_{0}\circ\gamma)(f_{1}\circ\gamma)^{\prime}\,dt$

for all $(f_{0},f_{1})\in{\mathscr{D}}^{1}(X)$. Similarly, every singular
Lipschitz $n$-chain in $X$ defines an element of ${\mathscr{D}}_{n}(X)$; in
fact, of ${\mathbf{I}}_{n,\text{\rm c}}(X)$ (see below for the definition, and
[15] for a reverse approximation result).

If $S\in{\mathscr{D}}_{n}(X)$ and $n\geq 1$, then the boundary $\partial
S\in{\mathscr{D}}_{n-1}(X)$ is defined by

$(\partial S)(f_{0},\dots,f_{n-1}):=S(\tau,f_{0},\dots,f_{n-1})$

for all $(f_{0},\ldots,f_{n-1})\in{\mathscr{D}}^{n-1}(X)$ and for any
$\tau\in{\mathscr{D}}^{0}(X)$ such that $\tau\equiv 1$ in a neighborhood of
$\operatorname{spt}(f_{0})$. It follows from (1) and (3) that $\partial S$ is
well-defined and that $\partial\circ\partial=0$. Furthermore,
$\operatorname{spt}(\partial S)\subset\operatorname{spt}(S)$, and
$\phi_{\\#}(\partial S)=\partial(\phi_{\\#}S)$ for $\phi\colon D\to Y$ as
above. In the example of a Lipschitz curve, $\partial(\gamma_{\\#}\llbracket
a,b\rrbracket)(f_{0})=f_{0}(\gamma(b))-f_{0}(\gamma(a))$ by the fundamental
theorem of calculus.

### Mass

Let $S\in{\mathscr{D}}_{n}(X)$. For an open set $U\subset X$, the mass
$\|S\|(U)\in[0,\infty]$ of $S$ in $U$ is defined as the supremum of
$\sum_{k}S(f_{0,k},\ldots,f_{n,k})$ over all finite families of tuples
$(f_{0,k},\ldots,f_{n,k})\in{\mathscr{D}}^{n}(X)$ such that
$\bigcup_{k}\operatorname{spt}(f_{0,k})\subset U$, $\sum_{k}|f_{0,k}|\leq 1$,
and $f_{1,k},\ldots,f_{n,k}$ are $1$-Lipschitz. This extends to a regular
Borel measure $\|S\|$ on $X$ with
$\operatorname{spt}(\|S\|)=\operatorname{spt}(S)$, and
${\mathbf{M}}(S):=\|S\|(X)$ denotes the total mass. For Borel sets
$W,A\subset{\mathbb{R}}^{n}$, $\|\llbracket W\rrbracket\|(A)$ equals the
Lebesgue measure of $W\cap A$. For $S,T\in{\mathscr{D}}_{n}(X)$,

$\|S+T\|\leq\|S\|+\|T\|.$

If the measure $\|S\|$ is locally finite, then

$|S(f_{0},\ldots,f_{n})|\leq\prod_{i=1}^{n}\operatorname{Lip}(f_{i})\int_{X}|f_{0}|\,d\|S\|$

for all $(f_{0},\ldots,f_{n})\in{\mathscr{D}}^{n}(X)$, where
$\operatorname{Lip}(f_{i})$ denotes the Lipschitz constant. As a consequence,
$S$ extends to tuples whose first entry is merely a bounded Borel function
with compact support, and if $u\colon X\to{\mathbb{R}}$ is any bounded Borel
function, one can define the restriction
$S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}u\in{\mathscr{D}}_{n}(X)$
by

$(S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}u)(f_{0},\dots,f_{n}):=S(uf_{0},f_{1},\ldots,f_{n})$

for all $(f_{0},\ldots,f_{n})\in{\mathscr{D}}^{n}(X)$. For a Borel set
$A\subset X$,
$S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}A:=S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}\chi_{A}$.
The measure
$\|S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}A\|$
agrees with the restriction of $\|S\|$ to $A$. If $\phi\colon D\to Y$ is as
above, and $B\subset Y$ is a Borel set, then
$(\phi_{\\#}S)\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B=\phi_{\\#}(S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}\phi^{-1}(B))$
and

$\|\phi_{\\#}S\|(B)\leq\operatorname{Lip}(\phi)^{n}\,\|S\|(\phi^{-1}(B)).$

### Integral currents

A current $S\in{\mathscr{D}}_{n}(X)$ is locally integer rectifiable if $\|S\|$
is locally finite and concentrated on the union of countably many Lipschitz
images of compact subsets of ${\mathbb{R}}^{n}$, and for every Borel set
$A\subset X$ with compact closure and every Lipschitz map
$\phi\colon\,\overline{\\!A}\to{\mathbb{R}}^{n}$, the current
$\phi_{\\#}(S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}A)\in{\mathscr{D}}_{n}({\mathbb{R}}^{n})$
is of the form $\llbracket w\rrbracket$ for some integer valued $w\in
L^{1}({\mathbb{R}}^{n})$. Then $\|S\|$ is absolutely continuous with respect
to $n$-dimensional Hausdorff measure. Push-forwards and restrictions to Borel
sets of locally integer rectifiable currents are again locally integer
rectifiable.

A current $S\in{\mathscr{D}}_{n}(X)$ is called a locally integral current if
$S$ is locally integer rectifiable and, for $n\geq 1$, $\|\partial S\|$ is
locally finite; then (by Theorem 8.7 in [25]) $\partial S$ is itself locally
integer rectifiable. This gives a chain complex of abelian groups
${\mathbf{I}}_{n,\text{\rm loc}}(X)$. The subgroups ${\mathbf{I}}_{n,\text{\rm
c}}(X)$ of integral currents consist of the elements with compact support and,
hence, finite total mass. For $X={\mathbb{R}}^{N}$, there is a canonical chain
isomorphism from ${\mathbf{I}}_{*,\text{\rm c}}({\mathbb{R}}^{N})$ to the
chain complex of classical (Federer–Fleming [13]) integral currents in
${\mathbb{R}}^{N}$.

For $n\geq 1$, we put ${\mathbf{Z}}_{n,\text{\rm
c}}(X):=\\{Z\in{\mathbf{I}}_{n,\text{\rm c}}(X):\partial Z=0\\}$. For $n=0$,
an element of ${\mathbf{I}}_{0,\text{\rm c}}(X)$ is an integral linear
combination of currents of the form $\llbracket x\rrbracket$, where
$\llbracket x\rrbracket(f_{0})=f_{0}(x)$. We let ${\mathbf{Z}}_{0,\text{\rm
c}}(X)\subset{\mathbf{I}}_{0,\text{\rm c}}(X)$ denote the subgroup of linear
combinations whose coefficients add up to zero. The boundary of a current in
${\mathbf{I}}_{1,\text{\rm c}}(X)$ belongs to ${\mathbf{Z}}_{0,\text{\rm
c}}(X)$. Given $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$, for $n\geq 0$, we will
call $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ a filling of $Z$ if $\partial
V=Z$.

### Slicing

Let $S\in{\mathbf{I}}_{n,\text{\rm loc}}(X)$, $n\geq 1$, and let $\pi\colon
X\to{\mathbb{R}}$ be a Lipschitz function. For $s\in{\mathbb{R}}$, the slice
$T_{s}\in{\mathscr{D}}_{n-1}(X)$ of $S$ with respect to $\pi$ is the current

$T_{s}:=\partial(S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}\\{\pi\leq
s\\})-(\partial
S)\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}\\{\pi\leq
s\\}$

with support in $\\{\pi=s\\}\cap\operatorname{spt}(S)$. Note that the
restrictions are defined since both $\|S\|$ and $\|\partial S\|$ are locally
finite. For $a<b$, the coarea inequality

$\int_{a}^{b}{\mathbf{M}}(T_{s})\,ds\leq\operatorname{Lip}(\pi)\,\|S\|(\\{a<\pi<b\\})$

holds, and if $\pi|_{\operatorname{spt}(S)}$ is proper, then
$T_{s}\in{\mathbf{I}}_{n-1,\text{\rm c}}(X)$ or almost all $s\in{\mathbb{R}}$.

### Convergence and compactness

A sequence $(S_{i})$ in ${\mathscr{D}}_{n}(X)$ converges weakly to a current
$S\in{\mathscr{D}}_{n}(X)$ if $S_{i}\to S$ pointwise as functionals on
${\mathscr{D}}^{n}(X)$. Then, for every open set $U\subset X$,

$\|S\|(U)\leq\liminf_{i\to\infty}\|S_{i}\|(U),$

thus the mass is lower semicontinuous with respect to weak convergence.
Furthermore, weak convergence commutes with the boundary operator and with
push-forwards. For locally integral currents, the following compactness
theorem holds (see Theorem 8.10 in [25]).

######  Theorem 2.1.

Let $X$ be a proper metric space, and let $n\geq 1$. If $(S_{i})$ is a
sequence in ${\mathbf{I}}_{n,\text{\rm loc}}(X)$ such that

$\sup_{i}(\|S_{i}\|+\|\partial S_{i}\|)(K)<\infty$

for every compact set $K\subset X$, then some subsequence $(S_{i_{k}})$
converges weakly to a current $S\in{\mathbf{I}}_{n,\text{\rm loc}}(X)$.

### Isoperimetric inequality and Plateau problem

Recall condition $({\rm CI}_{n})$ from the introduction. Cone inequalities are
instrumental for the proof of isoperimetric inequalities of Euclidean type
(compare Sect. 3.4 in [17]). For $n\geq 1$, we say that $X$ satisfies $({\rm
EII}_{n})$ if there is a constant $\gamma>0$ such that every cycle
$Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ has a filling
$V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ with mass

${\mathbf{M}}(V)\leq\gamma\,{\mathbf{M}}(Z)^{(n+1)/n}.$

To make the constants in $({\rm CI}_{n})$ and $({\rm EII}_{n})$ explicit, we
will write $({\rm CI}_{n})[c]$ and $({\rm EII}_{n})[\gamma]$. The following
result was established in a more general form in Theorem 1.2 in [31].

######  Theorem 2.2.

For all $n\geq 1$ and $c>0$ there is a constant $\gamma>0$ such that for every
proper metric space $X$, $({\rm CI}_{n})[c]$ implies $({\rm
EII}_{n})[\gamma]$.

(Here the quasi-convexity condition (CI0) is actually not needed.) By Theorem
2.1 and a well-known application of $({\rm EII}_{n})$ one gets the following
result (see the proof of Theorem 2.4 in [24]).

######  Theorem 2.3.

Let $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$, where $X$ is a proper metric space
satisfying $({\rm CI}_{0})$ if $n=0$ and $({\rm EII}_{n})[\gamma]$ if $n\geq
1$. Then there is a filling $V\in{\mathbf{I}}_{n+1,\text{\rm loc}}(X)$ of $Z$
with mass

${\mathbf{M}}(V)=\inf\\{{\mathbf{M}}(V^{\prime}):V^{\prime}\in{\mathbf{I}}_{n+1,\text{\rm
loc}}(X),\,\partial V^{\prime}=Z\\}<\infty.$

In fact, every such minimizing $V$ has compact support due to the following
lower density bound: if $x\in\operatorname{spt}(V)$, $r>0$, and
$B_{x}(r)\cap\operatorname{spt}(Z)=\emptyset$, then

$\Theta_{x,r}(V):=\frac{1}{r^{n+1}}\|V\|(B_{x}(r))\geq\delta_{0}:=\begin{cases}2&\text{if
$n=0$,}\\\ (n+1)^{-(n+1)}\gamma^{-n}&\text{if $n\geq 1$;}\end{cases}$

thus $\operatorname{spt}(V)$ is within distance
$({\mathbf{M}}(V)/\delta_{0})^{1/(n+1)}$ from $\operatorname{spt}(Z)$.

##  3\. A variational argument

We start with a slight modification and extension of an efficient recent
approximation result, Proposition 4.2 in [20]. The main conclusion is that for
a cycle $Z$ and any $\eta>0$ there is a cycle $Z^{\prime}$ with mass
$\leq{\mathbf{M}}(Z)$ such that $Z-Z^{\prime}$ has a filling with mass
$\leq\eta\,{\mathbf{M}}(Z)$ and $Z^{\prime}$ satisfies a uniform lower density
bound at scales $\lesssim\eta$. This will be used in the proofs of Theorem 3.2
and Theorem 5.1. We show in addition that if $Z$ satisfies a uniform upper
density bound above some threshold radius, then the same holds for
$Z^{\prime}$; see assertion (5) below. This will be employed in Theorem 5.3
(linear isoperimetric inequality).

######  Proposition 3.1.

Let $n\geq 1$ and $\gamma>0$. Suppose that $X$ is a proper metric space $X$
satisfying $({\rm EII}_{n})[\gamma]$ and, if $n\geq 2$, also $({\rm
EII}_{n-1})[\gamma]$. Then for every $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$
and $\eta>0$ there exists $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ such that
the following holds for

$Z^{\prime}:=Z-\partial V,\quad\mu:=\eta^{-1}\|V\|+\|Z^{\prime}\|,$

and some constants $\alpha,\theta>0$ depending only on $n$ and $\gamma$:

  1. (1)

$\mu(X)\leq{\mathbf{M}}(Z)$, in particular
${\mathbf{M}}(Z^{\prime})\leq{\mathbf{M}}(Z)$ and
${\mathbf{M}}(V)\leq\eta\,{\mathbf{M}}(Z)$;

  2. (2)

$\Theta_{x,r}(Z^{\prime})\geq\theta$ for all
$x\in\operatorname{spt}(Z^{\prime})$ and $r\in(0,\alpha\eta]$;

  3. (3)

if $B\subset X$ is a closed set and
$T:=\partial(V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B)-(\partial
V)\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B$ is
in ${\mathbf{I}}_{n,c}(X)$, then $\mu(B)\leq\|Z\|(B)+{\mathbf{M}}(T)$;

  4. (4)

if ${\mathbf{M}}(Z)<m:=\theta(\alpha\eta)^{n}$, then $Z^{\prime}=0$, and if
${\mathbf{M}}(Z)\geq m$, then $\operatorname{spt}(Z^{\prime})$ is within
distance at most $\eta(\alpha+\ln({\mathbf{M}}(Z)/m))$ from
$\operatorname{spt}(Z)$;

  5. (5)

if there exist $C>0$, $a\geq 0$, and $p\in X$ such that $\Theta_{p,r}(Z)\leq
C$ for all $r>a$, then $\mu(B_{p}(r))\leq 2^{n+1}Cr^{n}$ for all
$r>\max\\{a,2^{n+1}\eta\\}$.

###### Proof.

Given $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ and $\eta>0$, consider the
functional

$F\colon{\mathbf{I}}_{n+1,\text{\rm loc}}(X)\to[0,\infty],\quad
F(V^{\prime})=\eta^{-1}{\mathbf{M}}(V^{\prime})+{\mathbf{M}}(Z-\partial
V^{\prime}).$

Notice that $F$ is lower semicontinuous with respect to weak convergence, like
${\mathbf{M}}$. Moreover, ${\mathbf{M}}(V^{\prime})\leq\eta F(V^{\prime})$ and
${\mathbf{M}}(\partial V^{\prime})\leq F(V^{\prime})+{\mathbf{M}}(Z)$ for all
$V^{\prime}$, and $F(0)={\mathbf{M}}(Z)<\infty$. We can thus pick a minimizing
sequence for $F$ and use Theorem 2.1 to find a
$V\in{\mathbf{I}}_{n+1,\text{\rm loc}}(X)$ that minimizes $F$. Now if
$Z^{\prime}:=Z-\partial V$ and $\mu:=\eta^{-1}\|V\|+\|Z^{\prime}\|$, then

$\mu(X)=\eta^{-1}{\mathbf{M}}(V)+{\mathbf{M}}(Z^{\prime})=F(V)\leq
F(0)={\mathbf{M}}(Z),$

so (1) holds. However, we still have to show that in fact
$V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$.

We proceed with (2). Let $x\in\operatorname{spt}(Z^{\prime})$. Put
$f(s):=\|Z^{\prime}\|(B_{x}(s))>0$ for all $s>0$. For almost every $s$, the
slice
$R_{s}:=\partial(Z^{\prime}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(s))$
is in ${\mathbf{Z}}_{n-1,\text{\rm c}}(X)$ and satisfies
${\mathbf{M}}(R_{s})\leq f^{\prime}(s)$. Suppose first that $n\geq 2$. Then by
the isoperimetric inequality there exists a filling
$T_{s}\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ of $R_{s}$ such that

${\mathbf{M}}(T_{s})\leq\gamma\,{\mathbf{M}}(R_{s})^{n/(n-1)}\leq\gamma
f^{\prime}(s)^{n/(n-1)}.$

Furthermore, the cycle
$Z^{\prime}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(s)-T_{s}$
has a filling $W_{s}\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ with

${\mathbf{M}}(W_{s})\leq\gamma\bigl{(}f(s)+{\mathbf{M}}(T_{s})\bigr{)}^{(n+1)/n}.$

Since $Z-\partial(V+W_{s})=Z^{\prime}-\partial
W_{s}=Z^{\prime}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}(X\setminus
B_{x}(s))+T_{s}$, we have

$\displaystyle F(V+W_{s})$
$\displaystyle=\eta^{-1}{\mathbf{M}}(V+W_{s})+{\mathbf{M}}(Z^{\prime}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}(X\setminus
B_{x}(s))+T_{s})$
$\displaystyle\leq\eta^{-1}({\mathbf{M}}(V)+{\mathbf{M}}(W_{s}))+\|Z^{\prime}\|(X\setminus
B_{x}(s))+{\mathbf{M}}(T_{s}).$

It follows that $0\leq
F(V+W_{s})-F(V)\leq\eta^{-1}{\mathbf{M}}(W_{s})-f(s)+{\mathbf{M}}(T_{s})$ and

$\eta\bigl{(}f(s)-{\mathbf{M}}(T_{s})\bigr{)}\leq{\mathbf{M}}(W_{s})\leq\gamma\bigl{(}f(s)+{\mathbf{M}}(T_{s})\bigr{)}^{(n+1)/n}.$

Hence, if ${\mathbf{M}}(T_{s})\leq\frac{1}{2}f(s)$, then $\frac{1}{2}\eta
f(s)\leq\gamma(\frac{3}{2}f(s))^{(n+1)/n}$ and thus

$f(s)\geq\theta^{\prime}\eta^{n}$

for $\theta^{\prime}:=\frac{2}{3}(3\gamma)^{-n}$. Now if
$f(r)<\theta^{\prime}\eta^{n}$ for some $r$, then
$f(s)<\theta^{\prime}\eta^{n}$ for all $s\in(0,r)$, thus
$\frac{1}{2}f(s)<{\mathbf{M}}(T_{s})\leq\gamma f^{\prime}(s)^{n/(n-1)}$ and

$f^{\prime}(s)f(s)^{(1-n)/n}\geq(2\gamma)^{(1-n)/n}$

for almost every such $s$, and integration from $0$ to $r$ yields

$f(r)\geq\theta r^{n}$

where $\theta:=n^{-n}(2\gamma)^{1-n}$. This shows that
$f(r)\geq\min\\{\theta^{\prime}\eta^{n},\theta r^{n}\\}$ for all $r>0$. Hence
(2) holds with $\alpha:=(\theta^{\prime}/\theta)^{1/n}$ in case $n\geq 2$.
Suppose now that $n=1$. Then every non-zero slice
$R_{s}\in{\mathbf{Z}}_{0,\text{\rm c}}(X)$ has mass at least $2$. If $R_{s}=0$
for some $s$, then
$Z^{\prime}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(s)$
is a cycle, and repeating the above argument with $T_{s}=0$ we get that $\eta
f(s)\leq{\mathbf{M}}(W_{s})\leq\gamma f(s)^{2}$, thus $f(s)\geq\eta/\gamma$.
Hence, if $f(r)<\eta/\gamma$ for some $r$, then
$f^{\prime}(s)\geq{\mathbf{M}}(R_{s})\geq 2$ for almost every $s\in(0,r)$ and
so $f(r)\geq 2r$. We conclude that in case $n=1$, (2) holds with $\theta:=2$
and $\alpha:=1/(2\gamma)$.

Since ${\mathbf{M}}(Z^{\prime})<\infty$, it now follows from (2) that
$Z^{\prime}$ has compact support. Hence $\operatorname{spt}(\partial V)$ is
compact, and since $V$ has finite mass and is evidently minimizing, by Theorem
2.3 $\operatorname{spt}(V)$ is compact as well.

We prove (3). Let
$W:=V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B$.
If $T=\partial
W-(Z-Z^{\prime})\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$, then also
$(Z-Z^{\prime})\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$ and $W\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$. Since
$Z-\partial(V-W)=Z^{\prime}+\partial
W=Z^{\prime}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}(X\setminus
B)+Z\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B+T$,
we have

$\displaystyle F(V-W)$
$\displaystyle=\eta^{-1}{\mathbf{M}}(V-W)+{\mathbf{M}}\bigl{(}Z^{\prime}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}(X\setminus
B)+Z\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B+T\bigr{)}$
$\displaystyle\leq\mu(X\setminus B)+\|Z\|(B)+{\mathbf{M}}(T).$

Since $\mu(X)=F(V)\leq F(V-W)$, it follows that $\mu(B)=\mu(X)-\mu(X\setminus
B)\leq\|Z\|(B)+{\mathbf{M}}(T)$ as claimed.

The first assertion of (4) is clear from (1) and (2). Suppose now that
${\mathbf{M}}(Z)\geq m=\theta(\alpha\eta)^{n}$ and
$x\in\operatorname{spt}(Z^{\prime})$ is a point at distance $D>\alpha\eta$
from $\operatorname{spt}(Z)$. Set $g(s):=\mu(B_{x}(s))$ for all $s\in(0,D)$.
For almost every such $s$, the slice
$T_{s}:=\partial(V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(s))+Z^{\prime}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(s)$
is in ${\mathbf{I}}_{n,\text{\rm c}}(X)$ and satisfies
${\mathbf{M}}(T_{s})\leq\frac{d}{ds}\|V\|(B_{x}(s))$ as well as
${\mathbf{M}}(\partial T_{s})\leq\frac{d}{ds}\|Z^{\prime}\|(B_{x}(s))$. Hence,
by (3),

$g(s)\leq{\mathbf{M}}(T_{s})\leq{\mathbf{M}}(T_{s})+\eta\,{\mathbf{M}}(\partial
T_{s})\leq\eta\,g^{\prime}(s).$

Integrating the inequality $1\leq\eta\,g^{\prime}(s)/g(s)$ from $\alpha\eta$
to $t<D$ we get that

$t\leq\eta\bigl{(}\alpha+\ln(g(t))-\ln(g(\alpha\eta))\bigr{)}.$

By (1) and (2), $g(t)\leq\mu(X)\leq{\mathbf{M}}(Z)$ and
$g(\alpha\eta)\geq\|Z^{\prime}\|(B_{x}(\alpha\eta))\geq m$. As this holds for
all $t<D$, (4) follows.

It remains to prove (5). We will write $B_{r}$ for $B_{p}(r)$. First we choose
a sufficiently large $r_{0}>0$ so that

$\|V\|(B_{r_{0}})\leq 2^{n+1}\eta\,Cr_{0}^{\,n},$

then we put $r_{i}:=2^{-i}r_{0}$ for every integer $i\geq 1$. There exists an
$s\in(r_{1},r_{0})$ such that the slice
$T_{s}:=\partial(V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s})-(\partial
V)\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s}\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$ satisfies

$\mu(B_{s})\leq\|Z\|(B_{s})+{\mathbf{M}}(T_{s})$

by (3), as well as ${\mathbf{M}}(T_{s})\leq\|V\|(B_{r_{0}})/(r_{0}-r_{1})\leq
2^{n+1}\eta\,Cr_{0}^{\,n}/r_{1}$. Now if $r_{1}>\max\\{a,2^{n+1}\eta\\}$, then
$\|Z\|(B_{s})\leq Cs^{n}$ and ${\mathbf{M}}(T_{s})\leq Cr_{0}^{\,n}$, hence

$\mu(B_{r_{1}})\leq\mu(B_{s})\leq 2Cr_{0}^{\,n}=2^{n+1}Cr_{1}^{\,n}.$

This also yields the above inequality for the next smaller scale,

$\|V\|(B_{r_{1}})\leq 2^{n+1}\eta\,Cr_{1}^{\,n}.$

Finally, given any $r>\max\\{a,2^{n+1}\eta\\}$, we can choose $r_{0}$
initially such that $r=r_{k}=2^{-k}r_{0}$ for some $k\geq 1$. When $k\geq 2$,
we repeat the above slicing argument successively for $i=2,\dots,k$, with
$(r_{i},r_{i-1})$ in place of $(r_{1},r_{0})$. This eventually shows that

$\mu(B_{r})\leq 2^{n+1}Cr^{n}$

for any $r>\max\\{a,2^{n+1}\eta\\}$. ∎

As a first application of parts (1)–(3) of Proposition 3.1 we give a short
proof of a variant of Proposition 5.8 in [32] regarding fillings of thin
cycles. This result will play a key role in the next section (see Theorem
4.4). For $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ and $\epsilon>0$ we put

$\operatorname{FillVol}_{X}(Z):=\inf\\{{\mathbf{M}}(V):V\in{\mathbf{I}}_{n+1,\text{\rm
c}}(X),\,\partial V=Z\\}$

and $m_{\epsilon}(Z):=\sup_{x\in X}\|Z\|(B_{x}(\epsilon))$.

######  Theorem 3.2.

For all $n\geq 1$ and $\gamma,M,\epsilon,\nu>0$ there exists a constant
$\delta=\delta(n,\gamma,M,\epsilon,\nu)>0$ such that if $X$ is a proper metric
space satisfying $({\rm EII}_{n})[\gamma]$ and, in case $n\geq 2$, also $({\rm
EII}_{n-1})[\gamma]$, then every $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ with
${\mathbf{M}}(Z)\leq M$ and $m_{\epsilon}(Z)\leq\delta$ has
$\operatorname{FillVol}_{X}(Z)<\nu$.

###### Proof.

Let $\alpha$ and $\theta$ be the constants from Proposition 3.1, depending on
$n$ and $\gamma$. Given $M,\epsilon,\nu>0$, fix $\eta>0$ such that both $\eta
M$ and $\gamma(4\eta M/\epsilon)^{(n+1)/n}$ are less than $\frac{\nu}{2}$,
then put

$r:=\min\Bigl{\\{}\alpha\eta,\frac{\epsilon}{4}\Bigr{\\}},\quad\delta:=\frac{1}{2}\theta
r^{n}.$

Suppose now that $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ satisfies
${\mathbf{M}}(Z)\leq M$ and $m_{\epsilon}(Z)\leq\delta$. By Proposition 3.1
there exists $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ such that, for
$Z^{\prime}:=Z-\partial V$,

  1. (1)

${\mathbf{M}}(V)\leq\eta\,{\mathbf{M}}(Z)\leq\eta M<\frac{\nu}{2}$;

  2. (2)

$\|Z^{\prime}\|(B_{x}(r))\geq\theta r^{n}=2\delta$ for all
$x\in\operatorname{spt}(Z^{\prime})$;

  3. (3)

$\|Z^{\prime}\|(B)\leq\|Z\|(B)+{\mathbf{M}}(T)$ whenever $B\subset X$ is a
closed set and
$T:=\partial(V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B)-(\partial
V)\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B$ is
in ${\mathbf{I}}_{n,\text{\rm c}}(X)$.

If $Z^{\prime}=0$, then $\partial V=Z$, and (1) yields the result. Now let
$Z^{\prime}\neq 0$. It remains to show that
$\operatorname{FillVol}_{X}(Z^{\prime})<\frac{\nu}{2}$. Pick a maximal set
$N\subset\operatorname{spt}(Z^{\prime})$ of distinct points at mutual distance
$>2r$, and put $B_{s}:=\bigcup_{x\in N}B_{x}(s)$ for all $s>0$. By (2), since
the balls $B_{x}(r)$ with $x\in N$ are pairwise disjoint, we have
$2\delta\,|N|\leq\|Z^{\prime}\|(B_{r})\leq{\mathbf{M}}(Z^{\prime})$, and so
$\|Z\|(B_{\epsilon})\leq\delta\,|N|\leq\frac{1}{2}\,{\mathbf{M}}(Z^{\prime})$.
Furthermore, since $N$ is maximal, $\operatorname{spt}(Z^{\prime})\subset
B_{2r}\subset B_{\epsilon/2}$. Hence, for almost every
$s\in(\frac{\epsilon}{2},\epsilon)$, the slice
$T_{s}:=\partial(V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s})-(\partial
V)\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s}\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$ satisfies

${\mathbf{M}}(Z^{\prime})=\|Z^{\prime}\|(B_{s})\leq\|Z\|(B_{s})+{\mathbf{M}}(T_{s})\leq\frac{1}{2}\,{\mathbf{M}}(Z^{\prime})+{\mathbf{M}}(T_{s})$

by (3). We conclude that
${\mathbf{M}}(T_{s})\geq\frac{1}{2}\,{\mathbf{M}}(Z^{\prime})$ and

${\mathbf{M}}(V)\geq\|V\|(B_{\epsilon})\geq\int_{\epsilon/2}^{\epsilon}{\mathbf{M}}(T_{s})\,ds\geq\frac{\epsilon}{4}\,{\mathbf{M}}(Z^{\prime}).$

It follows from (1) that ${\mathbf{M}}(Z^{\prime})\leq 4\eta M/\epsilon$, and
by the isoperimetric inequality and the choice of $\eta$ we get that
$\operatorname{FillVol}_{X}(Z^{\prime})<\frac{\nu}{2}$. ∎

##  4\. Convergence of cycles

A central result in geometric measure theory says that a weakly convergent
sequence $S_{i}\to S$ of integral $n$-currents with supports in a fixed
compact set and with $\sup_{i}({\mathbf{M}}(S_{i})+{\mathbf{M}}(\partial
S_{i}))<\infty$ converges in the flat metric topology. This means that there
exist integral $n$-currents $T_{i}$ and integral $(n+1)$-currents $V_{i}$ such
that $S_{i}-S=T_{i}+\partial V_{i}$ and
${\mathbf{M}}(T_{i})+{\mathbf{M}}(V_{i})\to 0$. In ${\mathbb{R}}^{N}$, this
property can be deduced from the deformation theorem (see Theorem 5.5 and
Theorem 7.1 in [13]). The result was generalized in [32] to Ambrosio–Kirchheim
currents in complete metric spaces satisfying condition $({\rm CI}_{n})$
locally. It essentially suffices to show that
$\operatorname{FillVol}_{X}(Z_{i})\to 0$ for any bounded sequence of cycles
$Z_{i}$ converging weakly to $0$. In this section we prove a somewhat
amplified version of this fact for proper metric spaces satisfying $({\rm
CI}_{n})$ globally, so as to facilitate the proof of the sub-Euclidean
isoperimetric inequality in the next section. The argument relies on Theorem
3.2 and proceeds along the same lines as [32], but is simplified by the use of
a uniform notion of weak convergence.

Let $S\in{\mathscr{D}}_{n}(X)$, $n\geq 0$, and suppose that
$\operatorname{spt}(S)$ is compact. Note that every such $S$ extends
canonically to tuples of Lipschitz functions whose first entry is no longer
required to have compact support. We define

${\mathbf{W}}(S):=\sup\\{S(f_{0},\ldots,f_{n}):\text{$f_{0},\ldots,f_{n}$ are
$1$-Lipschitz, $|f_{0}|\leq 1$}\\}.$

Evidently ${\mathbf{W}}(S)\leq{\mathbf{M}}(S)$, and if $n\geq 1$, then
${\mathbf{W}}(\partial S)\leq{\mathbf{W}}(S)$.

The following auxiliary result is an adaptation of Proposition 6.6 in [25] to
sequences of cycles in possibly distinct proper metric spaces.

######  Lemma 4.1.

Suppose that $n\geq 1$, $Z_{i}\in{\mathbf{Z}}_{n,c}(X_{i})$,
$\sup_{i}{\mathbf{M}}(Z_{i})<\infty$, ${\mathbf{W}}(Z_{i})\to 0$, and
$\pi_{i}\colon X_{i}\to{\mathbb{R}}$ is $1$-Lipschitz. Then for almost every
$s\in{\mathbb{R}}$ there is a sequence $(i_{k})$ such that
$Z_{i_{k},s}:=Z_{i_{k}}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}\\{\pi_{i_{k}}\leq
s\\}\in{\mathbf{I}}_{n,\text{\rm c}}(X_{i_{k}})$ and

$\sup_{k}{\mathbf{M}}(\partial Z_{i_{k},s})<\infty,\quad{\mathbf{W}}(\partial
Z_{i_{k},s})\leq{\mathbf{W}}(Z_{i_{k},s})\to 0\quad(k\to\infty).$

###### Proof.

Consider the Borel measures $\mu_{i}:=\pi_{i\\#}\|Z_{i}\|$. Since
$\sup_{i}\mu_{i}({\mathbb{R}})<\infty$, some subsequence $(\mu_{i_{k}})$
converges weakly to a finite Borel measure $\mu$ on ${\mathbb{R}}$.
Furthermore, for the slices $\partial Z_{i_{k},s}$,

$\int_{{\mathbb{R}}}\liminf_{k\to\infty}{\mathbf{M}}(\partial
Z_{i_{k},s})\,ds\leq\liminf_{k\to\infty}\int_{{\mathbb{R}}}{\mathbf{M}}(\partial
Z_{i_{k},s})\,ds\leq\sup_{i}{\mathbf{M}}(Z_{i})<\infty.$

We now take $s$ so that $\mu(\\{s\\})=0$ and
$\liminf_{k\to\infty}{\mathbf{M}}(\partial Z_{i_{k},s})<\infty$, then we
adjust the sequence $(i_{k})$, if necessary, to arrange that
$\sup_{k}{\mathbf{M}}(\partial Z_{i_{k},s})<\infty$. Note that
$Z_{i_{k},s}\in{\mathbf{I}}_{n,\text{\rm c}}(X_{i_{k}})$ for all $k$. Let
$\epsilon>0$, and choose $\delta>0$ such that $\mu([s,s+\delta])<\epsilon$.
Let $\gamma_{s,\delta}\colon{\mathbb{R}}\to{\mathbb{R}}$ denote the piecewise
affine $\delta^{-1}$-Lipschitz function that is $1$ on $(-\infty,s]$ and $0$
on $[s+\delta,\infty)$, and put $u_{k}:=\gamma_{s,\delta}\circ\pi_{i_{k}}$ and
$v_{k}:=\chi_{(-\infty,s]}\circ\pi_{i_{k}}$. If $k$ is sufficiently large,
then

$\displaystyle{\mathbf{W}}(Z_{i_{k}}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}(u_{k}-v_{k}))\leq\mu_{i_{k}}([s,s+\delta])<\epsilon,$
$\displaystyle{\mathbf{W}}(Z_{i_{k}}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}u_{k})\leq(1+\delta^{-1}){\mathbf{W}}(Z_{i_{k}})<\epsilon$

(note that if $f_{0}\colon X_{i_{k}}\to{\mathbb{R}}$ is a $1$-Lipschitz
function with $|f_{0}|\leq 1$, then $u_{k}f_{0}$ is
$(1+\delta^{-1})$-Lipschitz), thus
${\mathbf{W}}(Z_{i_{k},s})={\mathbf{W}}(Z_{i_{k}}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}v_{k})<2\epsilon$.
This gives the result. ∎

Next, recall that a $0$-cycle $Z\in{\mathbf{Z}}_{0,\text{\rm c}}(X)$ is of the
form $Z(f)=\sum_{i}a_{i}f(x_{i})$ for finitely many points $x_{i}\in X$ and
weights $a_{i}\in{\mathbb{Z}}$ with $\sum_{i}a_{i}=0$. Note that $Z(f+c)=Z(f)$
for all $c\in{\mathbb{R}}$. We define

${\mathscr{W}}(Z):=\sup\\{Z(f):\text{$f$ is $1$-Lipschitz}\\};$

in general ${\mathscr{W}}(Z)\geq{\mathbf{W}}(Z)$. We have the following
optimal result for $n=0$ (compare Theorem 1.2 in [32]).

######  Proposition 4.2.

Let $X$ be a proper metric space satisfying $({\rm CI}_{0})[c]$, that is, any
two points $x,y\in X$ can be connected by a curve of length $\leq c\,d(x,y)$.
Then every cycle $Z\in{\mathbf{Z}}_{0,\text{\rm c}}(X)$ has
$\operatorname{FillVol}_{X}(Z)\leq c\,{\mathscr{W}}(Z)$, and if
${\mathbf{W}}(Z)<2$, then ${\mathscr{W}}(Z)={\mathbf{W}}(Z)$.

###### Proof.

We can write $Z\neq 0$ in the form

$Z(f)=\sum_{i=1}^{k}(f(x_{i})-f(y_{i}))=\int_{X}f\,d\mu-\int_{X}f\,d\nu$

for some (not necessarily distinct) points $x_{1},\ldots,x_{k}$ and
$y_{1},\ldots,y_{k}$ in $X$ and the corresponding measures
$\mu:=\sum_{i}\delta_{x_{i}}$ and $\nu:=\sum_{i}\delta_{y_{i}}$. Hence, by the
Kantorovich–Rubinstein theorem (see [11]), ${\mathscr{W}}(Z)$ equals the
Wasserstein distance $W_{1}(\mu,\nu)$. It is well-known that for such measures
the latter agrees with the minimum of $\sum_{i}d(x_{i},y_{\pi(i)})$ over all
permutations $\pi$ of $\\{1,\ldots,k\\}$. Thus, some such sum is equal to
${\mathscr{W}}(Z)$. We will give an alternative direct proof of this identity
in Lemma 4.3 below. It now follows from condition $({\rm CI}_{0})[c]$ that $Z$
has a filling with mass less than or equal to $c\,{\mathscr{W}}(Z)$.

For the second assertion of the proposition, consider the metric
$\delta:=\min\\{d,2\\}$ on $X$ and let ${\mathscr{W}}_{\delta}$ denote the
corresponding functional. Note that
${\mathscr{W}}_{\delta}(Z)={\mathbf{W}}(Z)$. If $Z$ is as above, then for some
$1$-Lipschitz function $f\colon(X,\delta)\to{\mathbb{R}}$ and some permutation
$\pi$, we have

${\mathscr{W}}_{\delta}(Z)=Z(f)=\sum_{i}(f(x_{i})-f(y_{\pi(i)}))=\sum_{i}\delta(x_{i},y_{\pi(i)}),$

in particular $f(x_{i})-f(y_{\pi(i)})=\delta(x_{i},y_{\pi(i)})\geq 0$ for all
$i$. We can assume that the set $\bigcup_{i}[f(y_{\pi(i)}),f(x_{i})]$ is
connected; otherwise $f$ can easily be modified so that this holds. Hence, if
${\mathscr{W}}_{\delta}(Z)={\mathbf{W}}(Z)<2$, then we can further arrange
that $|f|<1$. It follows that there is no $1$-Lipschitz function $g\colon
X\to{\mathbb{R}}$ with $Z(g)>{\mathbf{W}}(Z)=Z(f)$, for otherwise a suitable
convex combination $h=(1-\epsilon)f+\epsilon g$ would satisfy
$Z(h)>{\mathbf{W}}(Z)$ and $|h|\leq 1$. ∎

We now provide the alternative argument mentioned above. It is convenient to
consider pairwise distinct points but to allow distances to be zero.

######  Lemma 4.3.

Let $(V,d)$ be a finite pseudo-metric space with a partition $V=V_{+}\cup
V_{-}$, where $|V_{+}|=|V_{-}|$. If $f\colon V\to{\mathbb{R}}$ is a
$1$-Lipschitz function that maximizes the quantity $\sum_{x\in
V_{+}}f(x)-\sum_{y\in V_{-}}f(y)$, then there exists a bijection $\pi\colon
V_{+}\to V_{-}$ such that $f(x)-f(\pi(x))=d(x,\pi(x))$ for all $x\in V_{+}$.

###### Proof.

Given $f$, define a relation $\preceq$ on $V$ such that $x\preceq y$ if and
only if $f(x)-f(y)=d(x,y)$. Note that if $x\preceq y\preceq z$, then

$f(x)-f(z)=d(x,y)+d(y,z)\geq d(x,z),$

and since $f$ is $1$-Lipschitz, $x\preceq z$. Thus the relation is transitive.
For a set $A\subset V_{+}$, let $\Gamma(A)$ denote the set of all $y\in V_{-}$
for which there is an $x\in A$ with $x\preceq y$. Suppose first that $A$ is
maximal in $V_{+}$ in the sense that there is no pair $(x,y)\in
A\times(V_{+}\setminus A)$ with $x\preceq y$. Note that $f(x)-f(y)<d(x,y)$
whenever $x\in A$ and $y\in C:=(V_{+}\setminus
A)\cup(V_{-}\setminus\Gamma(A))$. By transitivity, the same strict inequality
holds whenever $x\in\Gamma(A)$ and $y\in C$. Hence, for some $\epsilon>0$, the
function $f_{\epsilon}$ obtained from $f$ by increasing the values on
$A\cup\Gamma(A)$ by $\epsilon$ is still $1$-Lipschitz. It follows from the
maximality property of $f$ that

$\epsilon\,|A|=\sum_{x\in V_{+}}(f_{\epsilon}(x)-f(x))\leq\sum_{y\in
V_{-}}(f_{\epsilon}(y)-f(y))=\epsilon\,|\Gamma(A)|,$

thus $|A|\leq|\Gamma(A)|$. Let now $A\subset V_{+}$ be arbitrary. Again by
transitivity, the set $A^{\prime}$ of all points in $V_{+}$ with a precursor
in $A$ is maximal, and $\Gamma(A^{\prime})=\Gamma(A)$, so that
$|A|\leq|A^{\prime}|\leq|\Gamma(A^{\prime})|=|\Gamma(A)|$. This shows that the
bipartite graph with edge set $\\{(x,y)\in V_{+}\times V_{-}:x\preceq y\\}$
satisfies the assumption of Hall’s marriage theorem. Hence, there is a
matching (bijection) $\pi$ as claimed. ∎

In general, for $n\geq 0$, an analog of Proposition 4.2 holds as follows.

######  Theorem 4.4.

If $X_{i}$ satisfies $({\rm CI}_{n})[c]$ for $i\in{\mathbb{N}}$, and if the
cycles $Z_{i}\in{\mathbf{Z}}_{n,c}(X_{i})$ satisfy
$\sup_{i}{\mathbf{M}}(Z_{i})<\infty$ and ${\mathbf{W}}(Z_{i})\to 0$, then
$\operatorname{FillVol}_{X_{i}}(Z_{i})\to 0$.

###### Proof.

The proof is by induction on $n$. For $n=0$, the result holds by Proposition
4.2. Assume now that $n\geq 1$ and the assertion holds in dimension $n-1$. It
suffices to show that for every sequence $(Z_{i})_{i\in{\mathbb{N}}}$ as in
the statement and for every $\nu>0$ there is an index $j$ with
$\operatorname{FillVol}_{X_{j}}(Z_{j})<\nu$. By Theorem 2.2 there is a
constant $\gamma=\gamma(n,c)$ such that every $X_{i}$ satisfies $({\rm
EII}_{n})[\gamma]$ and, if $n\geq 2$, also $({\rm EII}_{n-1})[\gamma]$. Put
$M:=\sup_{i}{\mathbf{M}}(Z_{i})$ and choose $\epsilon>0$ such that

$18c\epsilon M<\nu.$

Let $\delta:=\delta(n,\gamma,M,\epsilon,\frac{\nu}{2})$ be the constant from
Theorem 3.2. If there is an index $j$ with $m_{\epsilon}(Z_{j})\leq\delta$,
then $\operatorname{FillVol}_{X_{j}}(Z_{j})<\frac{\nu}{2}$ and we are done.

Suppose now that $m_{\epsilon}(Z_{i})>\delta$ for all $i$. Choose points
$x_{i}\in X_{i}$ such that

$\|Z_{i}\|(B_{x_{i}}(\epsilon))\geq\delta.$

By Lemma 4.1 there is an $s\in(\epsilon,2\epsilon)$ and an infinite set
$I_{1}\subset{\mathbb{N}}$ such that
$Z_{i,s}:=Z_{i}\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x_{i}}(s)\in{\mathbf{I}}_{n,\text{\rm
c}}(X_{i})$ for all $i\in I_{1}$ and

$\sup_{i\in I_{1}}{\mathbf{M}}(\partial
Z_{i,s})<\infty,\quad{\mathbf{W}}(\partial
Z_{i,s})\leq{\mathbf{W}}(Z_{i,s})\to 0\quad(I_{1}\ni i\to\infty).$

By the induction assumption there exist fillings
$T_{i}\in{\mathbf{I}}_{n,\text{\rm c}}(X_{i})$ of $\partial Z_{i,s}$ such that
${\mathbf{M}}(T_{i})\to 0$. By reducing the index set further, if necessary,
we arrange that ${\mathbf{M}}(T_{i})\leq\frac{\delta}{2}$ and
$\operatorname{spt}(T_{i})\subset B_{x_{i}}(3\epsilon)$ for all $i\in I_{1}$
(Theorem 2.3). Then $S_{i}^{1}:=Z_{i,s}-T_{i}$ is a cycle with support in
$B_{x_{i}}(3\epsilon)$, and

$\operatorname{FillVol}_{X_{i}}(S_{i}^{1})\leq
3c\epsilon\,{\mathbf{M}}(S_{i}^{1})$

by the cone inequality. Let $Z_{i}^{1}:=Z_{i}-Z_{i,s}+T_{i}$ and consider the
splitting

$Z_{i}=S_{i}^{1}+Z_{i}^{1}.$

Notice that
${\mathbf{M}}(S_{i}^{1})\leq{\mathbf{M}}(Z_{i,s})+\frac{\delta}{2}$ and
${\mathbf{M}}(Z_{i}^{1})\leq{\mathbf{M}}(Z_{i})-{\mathbf{M}}(Z_{i,s})+\frac{\delta}{2}$,
moreover ${\mathbf{M}}(Z_{i,s})\geq\|Z_{i}\|(B_{x_{i}}(\epsilon))\geq\delta$.
Hence, for all $i\in I_{1}$, we have

$\displaystyle{\mathbf{M}}(S_{i}^{1})$
$\displaystyle\leq{\mathbf{M}}(Z_{i})-{\mathbf{M}}(Z_{i}^{1})+\delta,$
$\displaystyle{\mathbf{M}}(Z_{i}^{1})$
$\displaystyle\leq{\mathbf{M}}(Z_{i})-\frac{\delta}{2}\leq M.$

Note further that
${\mathbf{W}}(Z_{i}^{1})\leq{\mathbf{W}}(Z_{i})+{\mathbf{W}}(Z_{i,s})+{\mathbf{M}}(T_{i})\to
0$ as $i\to\infty$.

If $m_{\epsilon}(Z_{i}^{1})>\delta$ for all $i\in I_{1}$, then we repeat the
above argument (with the same constants $M,\epsilon,\delta$) and produce
similar splittings $Z_{i}^{1}=S_{i}^{2}+Z_{i}^{2}$ for all $i$ in an infinite
set $I_{2}\subset I_{1}$. If $m_{\epsilon}(Z_{i}^{2})>\delta$ for all $i\in
I_{2}$, we iterate this step, and continue in this manner. This eventually
yields an infinite set $I_{k}\subset{\mathbb{N}}$, for some $k\geq 1$, and a
decomposition

$Z_{i}=S_{i}^{1}+\ldots+S_{i}^{k}+Z_{i}^{k}$

for every $i\in I_{k}$, such that $m_{\epsilon}(Z_{j}^{k})\leq\delta$ for some
$j\in I_{k}$. In fact, $k\leq 2M/\delta$, because
${\mathbf{M}}(Z_{i}^{k})\leq{\mathbf{M}}(Z_{i})-k\frac{\delta}{2}$ for all
$i\in I_{k}$. It follows that

$\displaystyle{\mathbf{M}}(S_{i}^{1})+\ldots+{\mathbf{M}}(S_{i}^{k})\leq{\mathbf{M}}(Z_{i})-{\mathbf{M}}(Z_{i}^{k})+k\delta\leq
3M,$
$\displaystyle\operatorname{FillVol}_{X_{i}}(S_{i}^{1})+\ldots+\operatorname{FillVol}_{X_{i}}(S_{i}^{k})\leq
9c\epsilon M<\frac{\nu}{2},$

and $\operatorname{FillVol}_{X_{j}}(Z_{j}^{k})<\frac{\nu}{2}$ by Theorem 3.2.
Hence, $\operatorname{FillVol}_{X_{j}}(Z_{j})<\nu$. ∎

The desired result for sequences of cycles with supports in a fixed compact
set now follows easily.

######  Theorem 4.5.

Let $X$ be a proper metric space satisfying condition $({\rm CI}_{n})$ for
some $n\geq 0$. If an ${\mathbf{M}}$-bounded sequence of cycles
$Z_{i}\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ with supports in a fixed compact
set $K\subset X$ converges weakly to $0$, then ${\mathbf{W}}(Z_{i})\to 0$ and
$\operatorname{FillVol}_{X}(Z_{i})\to 0$.

###### Proof.

Suppose that ${\mathbf{M}}(Z_{i})\leq M$ for all $i$. Let ${\mathscr{F}}$
denote the collection of all $1$-Lipschitz functions $f\colon
X\to{\mathbb{R}}$ with $|f|\leq\frac{1}{2}\operatorname{diam}(K)$. Let
$\epsilon>0$. There is a finite subcollection
${\mathscr{G}}\subset{\mathscr{F}}$ such that for all
$f_{0},\ldots,f_{n}\in{\mathscr{F}}$ there exist
$g_{0},\ldots,g_{n}\in{\mathscr{G}}$ with $\sup_{x\in
K}|f_{k}(x)-g_{k}(x)|\leq\epsilon/M$ for $k=0,\ldots,n$; then

$|Z_{i}(f_{0},\ldots,f_{n})-Z_{i}(g_{0},\ldots,g_{n})|\leq(n+1)\epsilon$

for all $i$ by Lemma 5.2 in [25]. As $Z_{i}\to 0$ weakly, if $i$ is
sufficiently large, then $Z_{i}(g_{0},\ldots,g_{n})\leq\epsilon$ for all
tuples $(g_{0},\ldots,g_{n})\in{\mathscr{G}}^{n+1}$, thus
$Z_{i}(f_{0},\ldots,f_{n})\leq(n+2)\epsilon$ whenever
$f_{0},\ldots,f_{n}\in{\mathscr{F}}$. Hence ${\mathbf{W}}(Z_{i})\to 0$, and
$\operatorname{FillVol}_{X}(Z_{i})\to 0$ by Theorem 4.4. ∎

##  5\. Isoperimetric inequalities

This section is devoted to isoperimetric inequalities and shows in particular
the implications $({\rm AR}_{n})\Rightarrow({\rm LII}_{n})\Leftrightarrow({\rm
SII}_{n})$ in Theorem 1.1. In fact we prove some stronger uniform statements.
To this end we first extend the notion of asymptotic rank to sequences of
metric spaces $X_{i}=(X_{i},d_{i})$. A compact metric space $\Omega$ is called
an asymptotic subset of the sequence $(X_{i})_{i\in{\mathbb{N}}}$ if there
exist positive numbers $r_{i}\to\infty$ and subsets $Y_{i}\subset X_{i}$ such
that the rescaled sets $(Y_{i},\frac{1}{r_{i}}d_{i})$ converge to $\Omega$ in
the Gromov–Hausdorff topology. The asymptotic rank of the sequence $(X_{i})$
is the supremum of all $k\geq 0$ such that there exists an asymptotic subset
bi-Lipschitz homeomorphic to a compact subset of ${\mathbb{R}}^{k}$ with
positive Lebesgue measure. The asymptotic rank of a single metric space $X$,
as defined in the introduction, equals the asymptotic rank of the constant
sequence $X_{i}=X$ (see Definition 1.1 and Proposition 3.1 in [34]).

From now on, throughout this section, we assume that ${\mathscr{X}}$ is a
class of proper metric spaces such that for some $n\geq 1$ and $c>0$, all
members of ${\mathscr{X}}$ satisfy $({\rm CI}_{n})[c]$, and every sequence
$(X_{i})_{i\in{\mathbb{N}}}$ in ${\mathscr{X}}$ has asymptotic rank $\leq n$.
The sub-Euclidean isoperimetric inequality for $n$-cycles in spaces of
asymptotic rank at most $n$ was established in greater generality in [34]
(Theorem 1.2), and a slightly restricted version was used as a key tool in
[24] (Theorem 4.4). First we give a proof of (a uniform variant of) the latter
statement.

######  Theorem 5.1.

For all $C,\epsilon>0$ there is a constant
$\varrho_{0}=\varrho_{0}({\mathscr{X}},n,c,C,\epsilon)>0$ such that if $X$
belongs to ${\mathscr{X}}$, and $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$
satisfies ${\mathbf{M}}(Z)\leq Cr^{n}$ and $\operatorname{spt}(Z)\subset
B_{p}(r)$ for some $p\in X$ and $r\geq\varrho_{0}$, then
$\operatorname{FillVol}_{X}(Z)<\epsilon r^{n+1}$.

###### Proof.

Suppose to the contrary that there exist $C,\epsilon>0$, a sequence of
positive radii $(r_{i})_{i\in{\mathbb{N}}}$ tending to infinity, and cycles
$Z_{i}\in{\mathbf{Z}}_{n,\text{\rm c}}(X_{i})$, where $X_{i}=(X_{i},d_{i})$
belongs to ${\mathscr{X}}$, each with mass ${\mathbf{M}}(Z_{i})\leq
Cr_{i}^{\,n}$ and support in some ball $B_{p_{i}}(r_{i})$, such that

$\operatorname{FillVol}_{X_{i}}(Z_{i})\geq\epsilon r_{i}^{\,n+1}.$

By Theorem 2.2 there is a constant $\gamma=\gamma(n,c)$ such that every
$X_{i}$ satisfies $({\rm EII}_{n})[\gamma]$ and, if $n\geq 2$, also $({\rm
EII}_{n-1})[\gamma]$. Put $\eta_{i}:=\epsilon r_{i}/(2C)$ and apply
Proposition 3.1 to $Z_{i}$ to get $V_{i}\in{\mathbf{I}}_{n+1,\text{\rm
c}}(X_{i})$ and $Z_{i}^{\prime}:=Z_{i}-\partial V_{i}$ such that

  1. (1)

${\mathbf{M}}(Z_{i}^{\prime})\leq Cr_{i}^{\,n}$ and
${\mathbf{M}}(V_{i})\leq\eta_{i}Cr_{i}^{\,n}=\epsilon r_{i}^{\,n+1}/2$;

  2. (2)

$\Theta_{x,r}(Z_{i}^{\prime})\geq\theta$ whenever
$x\in\operatorname{spt}(Z_{i}^{\prime})$ and
$0<r\leq\alpha\eta_{i}=\alpha\epsilon r_{i}/(2C)$, where $\alpha,\theta$
depend only on $n,\gamma$;

  3. (3)

$\operatorname{spt}(Z_{i}^{\prime})\subset B_{p_{i}}(\lambda r_{i})$ for some
constant $\lambda>1$ depending only on $n,\gamma,C,\epsilon$ (this uses part
(4) of Proposition 3.1).

By (1), (3) and the coning inequality $({\rm CI}_{n})[c]$ there exists a
filling $V_{i}^{\prime}\in{\mathbf{I}}_{n+1,\text{\rm c}}(X_{i})$ of
$Z_{i}^{\prime}$ with mass

${\mathbf{M}}(V_{i}^{\prime})\leq c\lambda
r_{i}\,{\mathbf{M}}(Z_{i}^{\prime})\leq c\lambda Cr_{i}^{\,n+1}.$

By Theorem 2.3 we can assume that $V_{i}^{\prime}$ is minimizing and has
support in $B_{p_{i}}(\lambda^{\prime}r_{i})$ for some
$\lambda^{\prime}>\lambda$ independent of $i$. Let $Y_{i}$ denote the set
$\operatorname{spt}(V_{i}^{\prime})$ equipped with the metric induced by
$\frac{1}{r_{i}}d_{i}$. Note that ${\mathbf{M}}(Z_{i}^{\prime})\leq C$ and
${\mathbf{M}}(V_{i}^{\prime})\leq c\lambda C$ with respect to this metric, and
$Y_{i}$ has diameter at most $2\lambda^{\prime}$. It follows from (2) and the
lower density bound for $V_{i}^{\prime}$ that the family of all $Y_{i}$ is
uniformly precompact. By Gromov’s compactness theorem [16], after passage to
subsequences and relabelling, there exist a compact metric space $Y$ and
isometric embeddings $\phi_{i}\colon Y_{i}\to Y$ such that the images
$\phi_{i}(Y_{i})$ converge to some compact set $\Omega\subset Y$ in the
Hausdorff distance. By Theorem 2.1 we can further assume that the push-
forwards $\phi_{i\\#}V_{i}^{\prime}\in{\mathbf{I}}_{n+1,\text{\rm c}}(Y)$
converge weakly to a current $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(Y)$.
Evidently $\operatorname{spt}(V)\subset\Omega$. Since the (sub)sequence
$(X_{i})_{i\in{\mathbb{N}}}$ has asymptotic rank $\leq n$, and $\Omega$ is an
asymptotic subset, it follows that there is no bi-Lipschitz embedding of a
compact subset of ${\mathbb{R}}^{n+1}$ with positive Lebesgue measure into
$\Omega$, and therefore $V$ must be zero (compare Theorem 8.3 in [25] and the
comments thereafter). Hence, the cycles
$\phi_{i\\#}Z_{i}^{\prime}\in{\mathbf{Z}}_{n,\text{\rm c}}(Y)$ converge weakly
to $\partial V=0$. We can assume that $Y$ satisfies condition $({\rm
CI}_{n})$; for example, the injective hull of $Y$ admits a conical geodesic
bicombing [26] and is still compact. It then follows from Theorem 4.5 that
${\mathbf{W}}(\phi_{i\\#}Z_{i}^{\prime})\to 0$. As this is an intrinsic
property of the currents, we conclude that ${\mathbf{W}}(Z_{i}^{\prime})\to 0$
with respect to the metrics $\frac{1}{r_{i}}d_{i}$. However, it follows from
the inequality $\operatorname{FillVol}_{X_{i}}(Z_{i})\geq\epsilon
r_{i}^{\,n+1}$ and (1) that
$\operatorname{FillVol}_{X_{i}}(Z_{i}^{\prime})\geq\frac{\epsilon}{2}$ with
respect to $\frac{1}{r_{i}}d_{i}$. This contradicts Theorem 4.4 (note that
$(X_{i},\frac{1}{r_{i}}d_{i})$ still satisfies $({\rm CI}_{n})[c]$). ∎

As a first application of Theorem 5.1 we derive a density bound for minimizing
fillings. This is similar to assertion (5) of Proposition 3.1 and to
Proposition 4.5 in [24].

######  Proposition 5.2.

For all $C,\delta>0$ there is a constant
$\varrho=\varrho({\mathscr{X}},n,c,C,\delta)>0$ such that if $X$ belongs to
${\mathscr{X}}$, and $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ is a cycle with
$\Theta_{p,r}(Z)\leq C$ for some $p\in X$ and for all $r>a\geq 0$, then every
minimizing filling $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ of $Z$ satisfies
$\Theta_{p,r}(V)<\delta$ for all $r>\max\\{\varrho,a\\}$.

###### Proof.

Let $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ be a minimizing filling of $Z$,
and set $B_{r}:=B_{p}(r)$ for all $r>0$. Choose a sufficiently large radius
$r_{0}>0$ such that

$\delta r_{0}^{\,n+1}>{\mathbf{M}}(V)\geq\|V\|(B_{r_{0}}),$

and put $r_{i}:=2^{-i}r_{0}$ for every integer $i\geq 1$. There exists an
$s\in(r_{1},r_{0})$ such that the slice
$T_{s}:=\partial(V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s})-Z\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s}$
is in ${\mathbf{I}}_{n,\text{\rm c}}(X)$ and has mass
${\mathbf{M}}(T_{s})\leq\|V\|(B_{r_{0}})/(r_{0}-r_{1})<2\delta r_{0}^{\,n}$.
Furthermore, if $r_{1}>a$, then
${\mathbf{M}}(Z\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s})\leq
Cs^{n}$ by assumption, thus the cycle
$Z_{s}:=Z\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s}+T_{s}$
satisfies

${\mathbf{M}}(Z_{s})\leq Cs^{n}+2\delta
r_{0}^{\,n}\leq(C+2\delta)r_{0}^{\,n},$

and $\operatorname{spt}(Z_{s})\subset B_{r_{0}}$. Note that
$V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s}$
is a minimizing filling of $Z_{s}$. By Theorem 5.1 there is a constant
$\varrho:=\frac{1}{2}\varrho_{0}({\mathscr{X}},n,c,C+2\delta,2^{-(n+1)}\delta)>0$
such that if $r_{1}>\max\\{\varrho,a\\}$ and, hence, $r_{0}\geq 2\varrho$,
then

$\|V\|(B_{r_{1}})\leq{\mathbf{M}}(V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{s})<2^{-(n+1)}\delta
r_{0}^{\,n+1}=\delta r_{1}^{\,n+1}.$

Finally, given any $r>\max\\{\varrho,a\\}$, we can choose $r_{0}$ initially
such that $r=r_{k}=2^{-k}r_{0}$ for some $k\geq 1$. When $k\geq 2$, we repeat
the above slicing argument successively for $i=2,\dots,k$, with
$(r_{i},r_{i-1})$ in place of $(r_{1},r_{0})$. This eventually shows that

$\|V\|(B_{r})<\delta r^{n+1}$

for all $r>\max\\{\varrho,a\\}$. ∎

Next we prove a linear isoperimetric inequality for cycles with controlled
density. This yields the implication $({\rm AR}_{n})\Rightarrow({\rm
LII}_{n})$ in Theorem 1.1.

######  Theorem 5.3 (linear isoperimetric inequality).

There is a constant $\nu=\nu(n,c)>0$, and for all $C>0$ there is a
$\lambda=\lambda({\mathscr{X}},n,c,C)>0$, such that if $X$ belongs to
${\mathscr{X}}$ and $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ is a cycle with
$(C,a)$-controlled density, $a\geq 0$, then
$\operatorname{FillVol}_{X}(Z)\leq\max\\{\lambda,\nu a\\}\,{\mathbf{M}}(Z)$.

###### Proof.

Note again that for some $\gamma=\gamma(n,c)$, every member of ${\mathscr{X}}$
satisfies $({\rm EII}_{n})[\gamma]$ and, if $n\geq 2$, also $({\rm
EII}_{n-1})[\gamma]$. Suppose that $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ has
$(C,a)$-controlled density. For any $\eta>0$, to be determined below,
Proposition 3.1 provides a $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ such that,
for $Z^{\prime}:=Z-\partial V\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ and some
constants $\alpha,\theta>0$ depending only on $n$ and $\gamma$,

  1. (1)

$\eta^{-1}{\mathbf{M}}(V)+{\mathbf{M}}(Z^{\prime})\leq{\mathbf{M}}(Z)$;

  2. (2)

$\Theta_{x,r}(Z^{\prime})\geq\theta$ for all
$x\in\operatorname{spt}(Z^{\prime})$ and $r\in(0,\alpha\eta]$;

  3. (3)

$Z^{\prime}$ has $(2^{n+1}C,\max\\{a,2^{n+1}\eta\\})$-controlled density.

By Theorem 2.3 there exists a minimizing filling
$V^{\prime}\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ of $Z^{\prime}$, and there
is a $\delta_{0}=\delta_{0}(n,\gamma)>0$ such that
$\Theta_{x,r}(V^{\prime})\geq\delta_{0}$ whenever
$x\in\operatorname{spt}(V^{\prime})$, $r>0$, and
$B_{x}(r)\cap\operatorname{spt}(Z^{\prime})=\emptyset$. On the other hand, by
(3) and Proposition 5.2, there is a constant
$\varrho^{\prime}:=\varrho({\mathscr{X}},n,c,2^{n+1}C,\delta_{0})>0$ such that
$\Theta_{p,r}(V^{\prime})<\delta_{0}$ for all $p\in X$ and
$r>\max\\{\varrho^{\prime},a,2^{n+1}\eta\\}$. We now fix $\eta$ so that

$2^{n+1}\eta=\max\\{\varrho^{\prime},a\\}.$

Then $\operatorname{spt}(V^{\prime})$ is within distance at most $2^{n+1}\eta$
from $\operatorname{spt}(Z^{\prime})$. Pick a maximal set
$N\subset\operatorname{spt}(Z^{\prime})$ of distinct points at mutual distance
$>2\alpha\eta$. The collection of all balls $B_{x}(2\alpha\eta)$ with $x\in N$
covers $\operatorname{spt}(Z^{\prime})$, and the corresponding balls with
radius

$r:=2(\alpha+2^{n})\eta$

cover $\operatorname{spt}(V^{\prime})$. Furthermore, the balls
$B_{x}(\alpha\eta)$ with $x\in N$ are pairwise disjoint, and
$\|Z^{\prime}\|(B_{x}(\alpha\eta))\geq\theta(\alpha\eta)^{n}$ by (2). Hence,
$|N|\leq{\mathbf{M}}(Z^{\prime})/(\theta\alpha^{n}\eta^{n})$, and since
$r>2^{n+1}\eta$, we have $\|V^{\prime}\|(B_{x}(r))<\delta_{0}r^{n+1}$ for all
$x\in N$. (Possibly $V^{\prime}=0$ and $|N|=0$.) Thus

${\mathbf{M}}(V^{\prime})\leq|N|\,\delta_{0}r^{n+1}\leq\frac{\delta_{0}r^{n+1}}{\theta\alpha^{n}\eta^{n}}\,{\mathbf{M}}(Z^{\prime})\leq\nu^{\prime}\eta\,{\mathbf{M}}(Z^{\prime})$

for some $\nu^{\prime}=\nu^{\prime}(n,c)\geq 1$. Now $V+V^{\prime}$ is a
filling of $Z$ with mass

${\mathbf{M}}(V+V^{\prime})\leq\nu^{\prime}({\mathbf{M}}(V)+\eta\,{\mathbf{M}}(Z^{\prime}))\leq\nu^{\prime}\eta\,{\mathbf{M}}(Z)$

by (1). In view of the choice of $\eta$, this gives the result. ∎

We now turn to the sub-Euclidean isoperimetric inequality as stated in Theorem
1.1. The proof below shows that $({\rm LII}_{n})\Rightarrow({\rm SII}_{n})$.

######  Theorem 5.4 (sub-Euclidean isoperimetric inequality).

For all $\epsilon>0$ there is a constant
$M_{0}=M_{0}({\mathscr{X}},n,c,\epsilon)>0$ such that if $X$ belongs to
${\mathscr{X}}$ and $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$, then
$\operatorname{FillVol}_{X}(Z)<\epsilon\max\\{M_{0},{\mathbf{M}}(Z)\\}^{(n+1)/n}$.

###### Proof.

Given $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$, note that if $t>0$ and
$r>t\,{\mathbf{M}}(Z)^{1/n}$, then

${\mathbf{M}}(Z)=t^{-n}\bigl{(}t\,{\mathbf{M}}(Z)^{1/n}\bigr{)}^{n}<t^{-n}r^{n},$

thus $Z$ has $(t^{-n},t\,{\mathbf{M}}(Z)^{1/n})$-controlled density. For
$\epsilon>0$, let $\nu=\nu(n,c)$ and $\lambda=\lambda({\mathscr{X}},n,c,C)$ be
the constants from Theorem 5.3, where now $C:=t^{-n}$ for any fixed
$t<\epsilon/\nu$. Let $M_{0}>0$ be such that $\lambda<\epsilon M_{0}^{1/n}$.
Then

$\max\\{\lambda,\nu
t\,{\mathbf{M}}(Z)^{1/n}\\}\,{\mathbf{M}}(Z)<\epsilon\max\\{M_{0},{\mathbf{M}}(Z)\\}^{(n+1)/n},$

and the result follows from Theorem 5.3. ∎

Finally, we show that Theorem 5.1 follows easily from Theorem 5.4. Given
$C,\epsilon>0$, put $\epsilon^{\prime}:=\epsilon/C^{(n+1)/n}$. If
$Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ is a cycle with ${\mathbf{M}}(Z)\leq
Cr^{n}$, and $r$ is sufficiently large, so that $Cr^{n}\geq
M_{0}=M_{0}(\epsilon^{\prime})$, then

$\operatorname{FillVol}_{X}(Z)<\epsilon^{\prime}(Cr^{n})^{(n+1)/n}=\epsilon
r^{n+1}$

by Theorem 5.4. Since the asymptotic rank assumption in Theorem 5.3 is only
used through Theorem 5.1, this also shows that $({\rm
SII}_{n})\Rightarrow({\rm LII}_{n})$.

##  6\. Quasiflats and quasi-minimizers

A map $f\colon W\to X$ from another metric space $W$ into $X$ is an
$(L,a)$-quasi-isometric embedding, for constants $L\geq 1$ and $a\geq 0$, if

$L^{-1}d(x,y)-a\leq d(f(x),f(y))\leq L\,d(x,y)+a$

for all $x,y\in W$. Propositions 3.6 and 3.7 in [24] show that quasi-isometric
embeddings of domains $W\subset{\mathbb{R}}^{n}$ into $X$ give rise to quasi-
minimizing currents with controlled density, as defined in the introduction.
An inspection of the proofs reveals that the statements hold in a stronger
form, in particular with a quasi-minimality constant $Q$ independent of the
parameter $a$. We provide the details for convenience, and also because parts
of the proof will be used later. The first result refers to the simpler case
when the map is actually Lipschitz.

######  Proposition 6.1.

For all $n\geq 1$ and $L\geq 1$ there exist $C>0$ and $Q\geq 1$ such that the
following holds. Let $W\subset{\mathbb{R}}^{n}$ be a compact set with finite
perimeter, so that the associated current $E:=\llbracket W\rrbracket$ (with
$\operatorname{spt}(E)\subset W$ and $\operatorname{spt}(\partial
E)\subset\partial W$) is in ${\mathbf{I}}_{n,\text{\rm c}}({\mathbb{R}}^{n})$.
Suppose that $a\geq 0$ and $f\colon W\to X$ is an $L$-Lipschitz,
$(L,a)$-quasi-isometric embedding into a proper metric space $X$. Then
$S:=f_{\\#}E\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ has $(C,a)$-controlled
density and is $(Q,Qa)$-quasi-minimizing mod $f(\partial W)$, furthermore
$d(f(x),\operatorname{spt}(S))\leq Qa$ for all $x\in W$ with $d(x,\partial
W)>Qa$.

###### Proof.

Let $B:=B_{p}(r)$ for some $p\in X$ and $r>a$. Then

$\|f_{\\#}E\|(B)\leq L^{n}\,\|E\|(f^{-1}(B)),$

and $f^{-1}(B)$ has diameter $\leq L(2r+a)\leq 3Lr$, thus $\|S\|(B)\leq
Cr^{n}$ for some constant $C=C(n,L)$. This shows that $S$ has
$(C,a)$-controlled density.

Next, let $V\subset W$ be a maximal subset of distinct points at mutual
distance $>2La$ ($V=W$ in case $a=0$). Note that $d(f(x),f(y))\geq
d(x,y)/(2L)$ for any $x,y\in V$, thus $f|_{V}$ has a $2L$-Lipschitz inverse,
which we can extend to an $\bar{L}$-Lipschitz map $\bar{f}\colon
X\to{\mathbb{R}}^{n}$ for some $\bar{L}=\bar{L}(n,L)$. Put $h:=\bar{f}\circ
f\colon W\to{\mathbb{R}}^{n}$. For every $x\in W$ there is a $y\in V$ with
$d(x,y)\leq 2La$; then $h(y)=y$ and

$d(h(x),x)\leq d(h(x),h(y))+d(y,x)\leq(\bar{L}L+1)\,d(x,y)\leq Na,$

where $N:=2(\bar{L}L+1)L$.

Let $x\in W$. Suppose that $r>2LNa$ and $B_{r}:=B_{f(x)}(r)$ is disjoint from
$f(\partial W)$. For almost every such $r$, both
$S_{r}:=S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{r}$
and
$E_{r}:=E\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}f^{-1}(B_{r})$
are integral currents, and $f_{\\#}E_{r}=S_{r}$. Since
$f^{-1}(B_{r})\cap\operatorname{spt}(\partial E)=\emptyset$, the support of
$\partial E_{r}$ lies in the boundary of $f^{-1}(B_{r})$ and is thus at
distance at least $L^{-1}r$ from $x$. The geodesic homotopy from the inclusion
map $W\to{\mathbb{R}}^{n}$ to $h$ provides a current
$R\in{\mathbf{I}}_{n,\text{\rm c}}({\mathbb{R}}^{n})$ with $\partial
R=h_{\\#}(\partial E_{r})-\partial E_{r}$ such that $\operatorname{spt}(R)$ is
within distance $Na$ from $\operatorname{spt}(\partial E_{r})$. In fact,
$R=\bar{f}_{\\#}S_{r}-E_{r}$, because $h_{\\#}(\partial
E_{r})=\partial(h_{\\#}E_{r})=\partial(\bar{f}_{\\#}S_{r})$ and
${\mathbf{Z}}_{n,\text{\rm c}}({\mathbb{R}}^{n})=\\{0\\}$. By the choice of
$r$ we have $\frac{r}{L}-Na>\frac{r}{2L}$, thus $\operatorname{spt}(R)$ lies
outside $B_{x}(\frac{r}{2L})$. It follows that

${\mathbf{M}}(\bar{f}_{\\#}S_{r})={\mathbf{M}}(E_{r}+R)\geq\|E\|(B_{x}(\tfrac{r}{2L}))\geq\epsilon
r^{n}$

for some $\epsilon=\epsilon(n,L)>0$. Now if $T\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$ is such that $\partial T=\partial S_{r}$, then
$\bar{f}_{\\#}T=\bar{f}_{\\#}S_{r}$, and

${\mathbf{M}}(S_{r})\leq Cr^{n}\leq
C\epsilon^{-1}{\mathbf{M}}(\bar{f}_{\\#}T)\leq Q^{\prime}\,{\mathbf{M}}(T)$

for $Q^{\prime}:=C\epsilon^{-1}\bar{L}^{n}$. This holds for all $x\in W$ and
almost all $r>2LNa$ as long as $B_{f(x)}(r)$ is disjoint from $f(\partial W)$.
In particular, since $\operatorname{spt}(S)\subset f(W)$, $S$ is
$(Q^{\prime},2LNa)$-quasi-minimizing mod $f(\partial W)$.

Finally, put $Q:=\max\\{Q^{\prime},L(2LN+1)\\}$. Let $x\in W$ with
$d(x,\partial W)>Qa$. Then $w:=d(f(x),f(\partial W))>L^{-1}Qa-a\geq 2LNa$. For
almost every $r\in(2LNa,w)$, the above argument shows that
${\mathbf{M}}(\bar{f}_{\\#}S_{r})>0$, thus
$S_{r}=S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{f(x)}(r)\neq
0$, and this implies that $d(f(x),\operatorname{spt}(S))\leq 2LNa\leq Qa$. ∎

For the second result, we suppose that $W$ is a triangulated polyhedral set,
that is, $W$ has the structure of a finite simplicial complex all of whose
maximal cells are Euclidean $n$-simplices. We write $W^{0}$ and $(\partial
W)^{0}$ for the set of vertices and boundary vertices of the triangulation,
respectively. Furthermore, $[\,\cdot\,]_{a}$ stands for the closed
$a$-neighborhood of a subset of $X$.

######  Proposition 6.2.

For all $n\geq 1$, $c>0$, and $K,L\geq 1$ there exist $C>0$ and $Q\geq 1$ such
that the following holds. Let $X$ be a proper metric space satisfying
condition $({\rm CI}_{n-1})[c]$. Suppose that $a>0$, and
$W\subset{\mathbb{R}}^{n}$ is a compact triangulated set with simplices of
diameter $\leq a$ such that every ball in ${\mathbb{R}}^{n}$ of radius $r>a$
intersects at most $Ka^{-n}r^{n}$ maximal simplices. Let
${\mathscr{P}}_{*}(W)$ denote the corresponding chain complex of simplicial
integral currents. If $f\colon W\to X$ is an $(L,a)$-quasi-isometric
embedding, then there exists a chain map
$\iota\colon{\mathscr{P}}_{*}(W)\to{\mathbf{I}}_{*,\text{\rm c}}(X)$ such that

  1. (1)

$\iota$ maps every vertex $\llbracket x_{0}\rrbracket\in{\mathscr{P}}_{0}(W)$
to $\llbracket f(x_{0})\rrbracket$ and, for $1\leq k\leq n$, every basic
oriented simplex $\llbracket
x_{0},\dots,x_{k}\rrbracket\in{\mathscr{P}}_{k}(W)$ to a minimizing current
with support in $[f(\\{x_{0},\dots,x_{k}\\})]_{Qa}$;

  2. (2)

$S:=\iota\llbracket W\rrbracket\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ has
$(C,a)$-controlled density and is $(Q,Qa)$-quasi-minimizing mod $[f((\partial
W)^{0})]_{Qa}$;

  3. (3)

$d(f(x),\operatorname{spt}(S))\leq Qa$ for all $x\in W$ with $d(x,(\partial
W)^{0})>Qa$.

Note that by (1), $\operatorname{spt}(S)\subset[f(W^{0})]_{Qa}$ and
$\operatorname{spt}(\partial S)\subset[f((\partial W)^{0})]_{Qa}$.

###### Proof.

Put ${\mathscr{S}}_{*}:=\bigcup_{k=0}^{n}{\mathscr{S}}_{k}$, where
${\mathscr{S}}_{k}$ denotes the set of all basic oriented simplices
$s=\llbracket x_{0},\dots,x_{k}\rrbracket\in{\mathscr{P}}_{k}(W)$ (compare p.
365 in [12] for the notation). We define a map
$\iota\colon{\mathscr{S}}_{*}\to{\mathbf{I}}_{*,\text{\rm c}}(X)$ by induction
on $k$. For $\llbracket x_{0}\rrbracket\in{\mathscr{S}}_{0}$, we put
$\iota\llbracket x_{0}\rrbracket:=f_{\\#}\llbracket x_{0}\rrbracket=\llbracket
f(x_{0})\rrbracket$. Suppose now that $k\in\\{1,\ldots,n\\}$ and $\iota$ is
defined on ${\mathscr{S}}_{k-1}$. For every $k$-cell of $W$, we choose an
orientation $s=\llbracket x_{0},\dots,x_{k}\rrbracket\in{\mathscr{S}}_{k}$,
then we let $\iota(s)\in{\mathbf{I}}_{k,\text{\rm c}}(X)$ be a minimizing
filling of the cycle

$\sum_{i=0}^{k}(-1)^{i}\,\iota\llbracket
x_{0},\dots,x_{i-1},x_{i+1},\dots,x_{k}\rrbracket\in{\mathbf{Z}}_{k-1,\text{\rm
c}}(X),$

and we put $\iota(-s):=-\iota(s)$. The resulting map on ${\mathscr{S}}_{*}$
readily extends to a chain map
$\iota\colon{\mathscr{P}}_{*}(W)\to{\mathbf{I}}_{*,\text{\rm c}}(X)$. Note
that $f$ maps the vertex set of any cell of $W$ to a set of diameter at most
$(L+1)a$. It follows inductively from condition $({\rm CI}_{n-1})[c]$ and
Theorem 2.3 (if $n\geq 2$, then $X$ satisfies $({\rm EII}_{n-1})$ by Theorem
2.2) that for all $s=\llbracket
x_{0},\dots,x_{k}\rrbracket\in{\mathscr{S}}_{k}$,

${\mathbf{M}}(\iota(s))\leq Ma^{k}$

and $\operatorname{spt}(\iota(s))\subset[f(\\{x_{0},\dots,x_{k}\\})]_{Ma}$ for
some constant $M\geq L+1$ depending only on $n,c,L$.

Let now ${\mathscr{S}}_{n}^{+}\subset{\mathscr{S}}_{n}$ be the set of all
positively oriented $n$-simplices, whose sum is $\llbracket W\rrbracket$. Put
$S:=\iota\llbracket W\rrbracket$. To show that $S$ has controlled density, let
$p\in X$ and $r>a$, and consider the set of all $s\in{\mathscr{S}}_{n}^{+}$
for which $\operatorname{spt}(\iota(s))\cap B_{p}(r)\neq\emptyset$. Every such
$s$ has a vertex $x^{s}$ with $f(x^{s})\in B_{p}(r+Ma)$, thus the set of all
$x^{s}$ has diameter at most $L(2(r+Ma)+a)\leq L(2M+3)r$. It follows that
there are at most $Ka^{-n}(L(2M+3)r)^{n}$ such simplices and that

$\Theta_{p,r}(S)\leq C:=KL^{n}(2M+3)^{n}M$

for $p\in X$ and $r>a$.

Similarly as in the proof of Proposition 6.1, there exists an
$\bar{L}$-Lipschitz map $\bar{f}\colon X\to{\mathbb{R}}^{n}$ such that
$h:=\bar{f}\circ f\colon W\to{\mathbb{R}}^{n}$ satisfies

$d(h(x),x)\leq Na$

for all $x\in W$, where $\bar{L}$ and $N$ depend only on $n,L$. Then

$\bar{\iota}:=\bar{f}_{\\#}\circ\iota\colon{\mathscr{P}}_{*}(W)\to{\mathbf{I}}_{*,\text{\rm
c}}({\mathbb{R}}^{n})$

is a chain map that sends every $\llbracket
x_{0}\rrbracket\in{\mathscr{S}}_{0}$ to $\llbracket h(x_{0})\rrbracket$ and
every $\llbracket x_{0},\dots,x_{k}\rrbracket\in{\mathscr{S}}_{k}$ to a
current with support in $[\\{x_{0},\dots,x_{k}\\}]_{(\bar{L}M+N)a}$. Let
${\mathscr{P}}_{*}(\partial W)$ be the complex of simplicial integral currents
in $\partial W$. A similar inductive construction as above, using minimizing
fillings of cycles in ${\mathbb{R}}^{n}$, produces a chain homotopy between
the inclusion map ${\mathscr{P}}_{*}(\partial W)\to{\mathbf{I}}_{*,\text{\rm
c}}({\mathbb{R}}^{n})$ and the restriction of $\bar{\iota}$ to
${\mathscr{P}}_{*}(\partial W)$. This yields an $R\in{\mathbf{I}}_{n,\text{\rm
c}}({\mathbb{R}}^{n})$ with boundary $\partial
R=\bar{\iota}(\partial\llbracket W\rrbracket)-\partial\llbracket W\rrbracket$
and support $\operatorname{spt}(R)\subset[(\partial W)^{0}]_{\bar{M}a}$ for
some constant $\bar{M}=\bar{M}(n,c,L)\geq 1$. In fact,

$R=\bar{f}_{\\#}S-\llbracket W\rrbracket,$

because $\bar{\iota}(\partial\llbracket
W\rrbracket)=\partial(\bar{\iota}\llbracket
W\rrbracket)=\partial(\bar{f}_{\\#}S)$.

Note that $\operatorname{spt}(S)\subset[f(W^{0})]_{Ma}$. Let
$\bar{x}\in[f(W^{0})]_{Ma}$ and $r>0$ be such that $B_{\bar{x}}(r)\cap
f((\partial W)^{0})=\emptyset$ and
$S_{r}:=S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{\bar{x}}(r)\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$. We want to show that if $r>Pa$, for some sufficiently large constant
$P=P(n,c,L)\geq 1$, then ${\mathbf{M}}(\bar{f}_{\\#}S_{r})\geq\epsilon r^{n}$
for some $\epsilon=\epsilon(n,L)>0$. Choose $x\in W^{0}$ with
$d(f(x),\bar{x})\leq Ma$, and put $B_{x}:=B_{x}(\tfrac{r}{2L})$. For all
$y\in(\partial W)^{0}$,

$r<d(\bar{x},f(y))\leq d(f(x),f(y))+Ma\leq L\,d(x,y)+(M+1)a$

and thus $d(x,y)>\frac{r}{2L}+\bar{M}a$ for sufficiently large $P$; then

$(\operatorname{spt}(R)\cup\partial W)\cap B_{x}=\emptyset.$

Moreover, for every
$\bar{y}\in\operatorname{spt}(S-S_{r})\subset\operatorname{spt}(S)$ there is a
vertex $y\in W^{0}$ such that $d(f(y),\bar{y})\leq Ma$,

$r\leq d(\bar{x},\bar{y})\leq d(f(x),f(y))+2Ma\leq L\,d(x,y)+(2M+1)a,$

and $d(x,y)\leq d(x,\bar{f}(\bar{y}))+d(\bar{f}(\bar{y}),h(y))+Na\leq
d(x,\bar{f}(\bar{y}))+(\bar{L}M+N)a$; thus
$d(x,\bar{f}(\bar{y}))>\frac{r}{2L}$ for sufficiently large $P$, implying that

$\operatorname{spt}(\bar{f}_{\\#}(S-S_{r}))\cap B_{x}=\emptyset.$

Since $\bar{f}_{\\#}S_{r}=\llbracket W\rrbracket+R-\bar{f}_{\\#}(S-S_{r})$, it
then follows that

${\mathbf{M}}(\bar{f}_{\\#}S_{r})\geq\|\llbracket
W\rrbracket\|(B_{x})\geq\epsilon r^{n}$

for some $\epsilon=\epsilon(n,L)>0$, as desired. Now if
$T\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ is such that $\partial T=\partial
S_{r}$, then $\bar{f}_{\\#}T=\bar{f}_{\\#}S_{r}$, and

${\mathbf{M}}(S_{r})\leq Cr^{n}\leq
C\epsilon^{-1}{\mathbf{M}}(\bar{f}_{\\#}T)\leq Q^{\prime}\,{\mathbf{M}}(T)$

for $Q^{\prime}:=C\epsilon^{-1}\bar{L}^{n}$. Since $\operatorname{spt}(S)$ and
$\operatorname{spt}(\partial S)$ are within distance $Ma$ from $f(W^{0})$ and
$f((\partial W)^{0})$, respectively, this shows in particular that $S$ is
$(Q^{\prime},Pa)$-quasi-minimizing mod $[f((\partial W)^{0})]_{Ma}$.

Finally, put $Q:=\max\\{M,Q^{\prime},L(P+1)\\}$. Let $x^{\prime}\in W$ with
$d(x^{\prime},(\partial W)^{0})>Qa$. Then $w:=d(f(x^{\prime}),f((\partial
W)^{0}))>L^{-1}Qa-a\geq Pa$. Note that $f(x^{\prime})\in[f(W^{0})]_{Ma}$, as
$M\geq L+1$. For $\bar{x}=f(x^{\prime})$ and almost every $r\in(Pa,w)$, the
above argument shows that ${\mathbf{M}}(\bar{f}_{\\#}S_{r})>0$, thus
$S_{r}=S\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{\bar{x}}(r)\neq
0$, and this implies that $d(f(x^{\prime}),\operatorname{spt}(S))\leq Pa\leq
Qa$. ∎

##  7\. Morse lemma, slim simplices, and filling radius

We now turn to the remaining assertions in Theorem 1.1. For the first three
results, we assume as in Sect. 5 that ${\mathscr{X}}$ is a class of proper
metric spaces such that for some $n\geq 1$ and $c>0$, all members of
${\mathscr{X}}$ satisfy condition $({\rm CI}_{n})[c]$, and every sequence
$(X_{i})_{i\in{\mathbb{N}}}$ in ${\mathscr{X}}$ has asymptotic rank $\leq n$.
We begin with a uniform version of the Morse lemma, analogous to Theorem 5.1
in [24]. The asymptotic rank assumption is only used through Proposition 5.2,
or Theorem 5.1, which in turn follows from Theorem 5.4. Hence, $({\rm
SII}_{n})\Rightarrow({\rm ML}_{n})$.

######  Theorem 7.1 (Morse lemma).

For all $C>0$ and $Q\geq 1$ there is a constant
$l=l({\mathscr{X}},n,c,C,Q)\geq 0$ such that if $X$ belongs to
${\mathscr{X}}$, and $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ has
$(C,a)$-controlled density and is $(Q,a)$-quasi-minimizing mod $Y$, where
$Y\subset X$ is a closed set and $a\geq 0$, then the support of $Z$ is within
distance at most $\max\\{l,4a\\}$ from $Y$.

###### Proof.

Let $x\in\operatorname{spt}(Z)\setminus Y$. Essentially the same argument as
for the second part of Theorem 2.3 (using $({\rm EII}_{n-1})$ if $n\geq 2$)
shows that there is a constant
$\delta_{0}^{\prime}=\delta_{0}^{\prime}(n,c)>0$ such that
$\Theta_{x,s}(Z)\geq\delta_{0}^{\prime}\,Q^{1-n}$ whenever $s>2a$ and
$B_{x}(s)\cap Y=\emptyset$ (see Lemma 3.3 in [24]). Now let
$V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ be a minimizing filling of $Z$, and
suppose that $r>4a$ and $B_{x}(r)\cap Y=\emptyset$. For almost every
$s\in(2a,r)$, the slice
$T_{s}=\partial(V\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(s))-Z\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(s)\in{\mathbf{I}}_{n,\text{\rm
c}}(X)$ satisfies

$Q\,{\mathbf{M}}(T_{s})\geq{\mathbf{M}}(Z\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}B_{x}(s))\geq\delta_{0}^{\prime}\,Q^{1-n}s^{n},$

and integrating the inequality
${\mathbf{M}}(T_{s})\geq\delta_{0}^{\prime}\,Q^{-n}s^{n}$ from
$\frac{r}{2}>2a$ to $r$ we get that $\Theta_{x,r}(V)\geq\delta$ for some
$\delta=\delta(n,c,Q)>0$ (compare Lemma 3.4 in [24]). On the other hand, by
Proposition 5.2 there is a constant $l:=\varrho({\mathscr{X}},n,c,C,\delta)>0$
such that $\Theta_{x,r}(V)<\delta$ for all $r>\max\\{l,a\\}$. Hence,
$r\leq\max\\{l,4a\\}$. ∎

The next statement strengthens Theorem 5.2 in [24]. The proof shows that
$({\rm ML}_{n})\Rightarrow(\SS_{n})$. A facet of an $(n+1)$-simplex is an
$n$-dimensional face.

######  Theorem 7.2 (slim simplices).

For all $L\geq 1$ there is a constant $D=D({\mathscr{X}},n,c,L)\geq 0$ such
that the following holds. Let $\Delta$ be a Euclidean $(n+1)$-simplex, $X$ a
member of ${\mathscr{X}}$, and $a\geq 0$. Suppose that
$f\colon\partial\Delta\to X$ is a map whose restriction to each facet of
$\Delta$ is an $(L,a)$-quasi-isometric embedding. Then the image of every
facet is within distance at most $D(1+a)$ from the union of the images of the
remaining ones.

###### Proof.

Let $W_{0},\dots,W_{n+1}\subset\partial\Delta$ be an enumeration of the
(closed) facets of $\Delta$, and let
$E_{i}:=(\partial\llbracket\Delta\rrbracket)\mathop{\mbox{\rule{0.43057pt}{5.16663pt}\rule{4.73611pt}{0.43057pt}}}W_{i}\in{\mathbf{I}}_{n,\text{\rm
c}}({\mathbb{R}}^{n+1})$ denote the corresponding currents, whose sum is the
boundary cycle $\partial\llbracket\Delta\rrbracket\in{\mathbf{Z}}_{n,\text{\rm
c}}({\mathbb{R}}^{n+1})$.

Suppose that $a>0$. Choose a triangulation of $\partial\Delta$ with simplices
of diameter $\leq a$ such that, for some constant $K=K(n)$ and for each $i$,
any ball in ${\mathbb{R}}^{n+1}$ of radius $r>a$ intersects at most
$Ka^{-n}r^{n}$ maximal simplices in $W_{i}$. Let
${\mathscr{P}}_{*}(\partial\Delta)$ be the corresponding chain complex of
simplicial integral currents. A slight adaptation of Proposition 6.2 provides
a chain map
$\iota\colon{\mathscr{P}}_{*}(\partial\Delta)\to{\mathbf{I}}_{*,\text{\rm
c}}(X)$ such that the following properties hold for each
$S_{i}:=\iota(E_{i})\in{\mathbf{I}}_{n,\text{\rm c}}(X)$ and for some
constants $C,Q$ depending only on $n,c,L$:

  1. (1)

$\operatorname{spt}(S_{i})\subset[f(W_{i})]_{Qa}$ and
$\operatorname{spt}(\partial S_{i})\subset[f(\partial W_{i})]_{Qa}$;

  2. (2)

$S_{i}$ has $(C,a)$-controlled density and is $(Q,Qa)$-quasi-minimizing mod
$[f(\partial W_{i})]_{Qa}$;

  3. (3)

$d(f(x),\operatorname{spt}(S_{i}))\leq Qa$ for all $x\in W_{i}$ with
$d(x,\partial W_{i})>Qa$.

Here $[\,\cdot\,]_{Qa}$ stands again for the closed $Qa$-neighborhood, and
$\partial W_{i}$ denotes the relative boundary of $W_{i}$. Let $M_{i}$ denote
the union of all $W_{j}$ with $j\neq i$. The cycle
$Z:=\iota(\partial\llbracket\Delta\rrbracket)=\sum_{i=0}^{n+1}S_{i}$ has
$((n+2)C,a)$-controlled density and is $(Q,Qa)$-quasi-minimizing mod
$[f(M_{i})]_{Qa}$ for every $i$. It then follows from Theorem 7.1 that the set
$\operatorname{spt}(S_{i})\setminus[f(M_{i}))]_{Qa}=\operatorname{spt}(Z)\setminus[f(M_{i})]_{Qa}$
is within distance at most $\max\\{l^{\prime},4Qa\\}$ from $[f(M_{i})]_{Qa}$
for some $l^{\prime}=l^{\prime}({\mathscr{X}},n,c,L)$. Hence, for any $x\in
W_{i}$, it follows from (3) that $d(f(x),f(M_{i}))$ is less than or equal to
$2Qa+\max\\{l^{\prime},4Qa\\}$ if $d(x,\partial W_{i})>Qa$ and less than or
equal to $LQa+a$ otherwise.

Note that if the restriction of $f$ to each facet of $\Delta$ is $L$-Lipschitz
in addition, or if $a=0$, then the proof can be simplified by using
Proposition 6.1 instead of Proposition 6.2. ∎

The proof of the following result relies again on Proposition 5.2; thus $({\rm
SII}_{n})\Rightarrow({\rm FR}_{n})$.

######  Theorem 7.3 (filling radius).

For all $C>0$ there is a constant $h=h({\mathscr{X}},n,c,C)>0$ such that if
$X$ belongs to ${\mathscr{X}}$ and $Z\in{\mathbf{Z}}_{n,\text{\rm c}}(X)$ has
$(C,a)$-controlled density for some $a\geq 0$, then the support of every
minimizing filling $V\in{\mathbf{I}}_{n+1,\text{\rm c}}(X)$ of $Z$ is within
distance at most $\max\\{h,a\\}$ from $\operatorname{spt}(Z)$.

###### Proof.

Suppose that $x\in\operatorname{spt}(V)\setminus\operatorname{spt}(Z)$. By
Theorem 2.2 and Theorem 2.3 there are constants $\gamma=\gamma(n,c)$ and
$\delta_{0}=\delta_{0}(n,\gamma)>0$ such that $\Theta_{x,r}(V)\geq\delta_{0}$
whenever $r>0$ and $B_{x}(r)\cap\operatorname{spt}(Z)=\emptyset$. On the other
hand, Proposition 5.2 shows that there is a constant
$h=\varrho({\mathscr{X}},n,c,C,\delta_{0})>0$ such that
$\Theta_{x,r}(V)<\delta_{0}$ for all $r>\max\\{h,a\\}$. Thus there is no point
$x\in\operatorname{spt}(V)$ at distance bigger than $\max\\{h,a\\}$ from
$\operatorname{spt}(Z)$. ∎

We now prove the implication $(\SS_{n})\Rightarrow({\rm AR}_{n})$, which holds
without further assumptions on the metric space $X$.

######  Proposition 7.4.

Let $(X_{i})_{i\in{\mathbb{N}}}$ be a sequence of metric spaces
$X_{i}=(X_{i},d_{i})$, let $n\geq 1$, and suppose that for every $L\geq 1$
there exists $D\geq 0$ such that every $X_{i}$ satisfies $(\SS_{n})$ with
constant $D=D(L)$. Then the sequence $(X_{i})_{i\in{\mathbb{N}}}$ has
asymptotic rank $\leq n$.

###### Proof.

Suppose to the contrary that the sequence $(X_{i})_{i\in{\mathbb{N}}}$ has
asymptotic rank $>n$. Then there exist a compact set
$K\subset{\mathbb{R}}^{n+1}$ with positive Lebesgue measure, an $L$-bi-
Lipschitz map $\phi\colon K\to\Omega$ onto some metric space $\Omega$, and a
sequence of $(1,\delta_{i})$-quasi-isometric embeddings
$h_{i}\colon\Omega\to(X_{i},\frac{1}{r_{i}}d_{i})$, where $L\geq 1$,
$\delta_{i}\to 0$, and $r_{i}\to\infty$. We can assume that
$0\in{\mathbb{R}}^{n+1}$ is a Lebesgue density point of $K$. Let
$B:=B_{0}(1)\subset{\mathbb{R}}^{n+1}$. For all $k\in{\mathbb{N}}$ there is a
$\lambda_{k}>0$ such that every point in $\lambda_{k}B$ is at distance
$\leq\frac{1}{2k}{\lambda_{k}}$ from some point in $K$, thus there exist
$(1,\frac{1}{k}\lambda_{k})$-quasi-isometric embeddings
$\psi_{k}\colon\lambda_{k}B\to K$. Choose $i(k)\in{\mathbb{N}}$ such that
$s_{k}:=\lambda_{k}r_{i(k)}\to\infty$ and
$\epsilon_{k}:=\frac{1}{\lambda_{k}}\delta_{i(k)}+\frac{1}{k}L\to 0$. It is
straightforward to check that the map

$f_{k}\colon s_{k}B\to(X_{i(k)},d_{i(k)})$

defined by $f_{k}(s_{k}x)=h_{i(k)}\circ\phi\circ\psi_{k}(\lambda_{k}x)$ for
all $x\in B$ is an $(L,\epsilon_{k}s_{k})$-quasi-isometric embedding.

Now let $\Delta$ be any $(n+1)$-simplex inscribed in $B$, and pick a point $x$
in a facet of $\Delta$ such that $x$ is a distance $\delta>0$ away from the
union $M$ of the remaining facets. For every $k$, the point $f_{k}(s_{k}x)$ is
at distance at least $L^{-1}\delta s_{k}-\epsilon_{k}s_{k}$ from
$f_{k}(s_{k}M)$. On the other hand, by assumption, there is a constant
$D=D(L)$ such that the distance is at most $D(1+\epsilon_{k}s_{k})$. This
leads to the inequality $L^{-1}\delta-\epsilon_{k}\leq
D(\frac{1}{s_{k}}+\epsilon_{k})$, which contradicts the fact that
$s_{k}\to\infty$ and $\epsilon_{k}\to 0$. ∎

Lastly, we show that $({\rm FR}_{n})\Rightarrow({\rm AR}_{n})$. This is
similar to Theorem 6.1 in [34].

######  Proposition 7.5.

Let $n\geq 1$ and $c>0$, and let $(X_{i})_{i\in{\mathbb{N}}}$ be a sequence of
proper metric spaces $X_{i}=(X_{i},d_{i})$ satisfying $({\rm CI}_{n})[c]$.
Suppose further that for every $C>0$ there exists $h>0$ such that every
$X_{i}$ satisfies $({\rm FR}_{n})$ with constant $h=h(C)$. Then the sequence
$(X_{i})_{i\in{\mathbb{N}}}$ has asymptotic rank $\leq n$.

###### Proof.

Suppose to the contrary that $(X_{i})_{i\in{\mathbb{N}}}$ has asymptotic rank
$>n$. Let $L,s_{k},\epsilon_{k}$ and $f_{k}\colon s_{k}B\to X_{i(k)}$ be given
as in the first part of the proof of Proposition 7.4. Let again $\Delta$ be
any $(n+1)$-simplex inscribed in $B$. For every $k$, fix a triangulation of
$\partial\Delta$ with simplices of diameter at most $\epsilon_{k}$ such that,
for some constant $K=K(n)$, every ball in ${\mathbb{R}}^{n+1}$ of radius
$r>\epsilon_{k}$ meets at most $K(r/\epsilon_{k})^{n}$ maximal simplices in
each facet of $\Delta$. It then follows as in the proof of Proposition 6.2
that for every $k$, and for some constants $C,\bar{L},\bar{M}$ depending only
on $n,c,L$, there exist a cycle $Z_{k}\in{\mathbf{Z}}_{n,\text{\rm
c}}(X_{i(k)})$ with $(C,\epsilon_{k}s_{k})$-controlled density, an
$\bar{L}$-Lipschitz map $\bar{f}_{k}\colon X_{i(k)}\to{\mathbb{R}}^{n+1}$, and
a current $R_{k}\in{\mathbf{I}}_{n+1,\text{\rm c}}({\mathbb{R}}^{n+1})$ such
that $\partial R_{k}=\bar{f}_{k\\#}Z_{k}-\partial\llbracket
s_{k}\Delta\rrbracket$ and

$\operatorname{spt}(R_{k})\cup\bar{f}_{k}(\operatorname{spt}(Z_{k}))\subset[\partial(s_{k}\Delta)]_{\bar{M}\epsilon_{k}s_{k}}.$

Fix a point $x\in\Delta$ a distance $\delta>0$ away from $\partial\Delta$.
Suppose that $k$ is so large that $\bar{M}\epsilon_{k}<\delta$, and
$V_{k}\in{\mathbf{I}}_{n+1,\text{\rm c}}(X_{i(k)})$ is any filling of $Z_{k}$.
Then $\partial\llbracket
s_{k}\Delta\rrbracket=\partial(\bar{f}_{k\\#}V_{k})-\partial R_{k}$, hence
$\llbracket s_{k}\Delta\rrbracket=\bar{f}_{k\\#}V_{k}-R_{k}$ and
$s_{k}\Delta\subset\operatorname{spt}(\bar{f}_{k\\#}V_{k})\cup\operatorname{spt}(R_{k})$.
Since $s_{k}x\not\in\operatorname{spt}(R_{k})$, there is a point
$y_{k}\in\operatorname{spt}(V_{k})$ such that $\bar{f}_{k}(y_{k})=s_{k}x$. For
every $z_{k}\in\operatorname{spt}(Z_{k})$, we have

$s_{k}\delta=d(s_{k}x,s_{k}\Delta)\leq
d(s_{k}x,\bar{f}_{k}(z_{k}))+\bar{M}\epsilon_{k}s_{k}\leq\bar{L}\,d_{k}(y_{k},z_{k})+\bar{M}\epsilon_{k}s_{k}.$

On the other hand, by assumption, there exists a filling $V_{k}$ of $Z_{k}$
whose support is within distance $\max\\{h,\epsilon_{k}s_{k}\\}$ from
$\operatorname{spt}(Z_{k})$, where $h=h(C)$. This leads to the inequality
$\delta\leq\bar{L}\max\\{\frac{1}{s_{k}}h,\epsilon_{k}\\}+\bar{M}\epsilon_{k}$,
which contradicts the fact that $s_{k}\to\infty$ and $\epsilon_{k}\to 0$. ∎

The uniform statements in Sect. 5 and above can be combined to show that the
implications in Theorem 1.1 that we proved through $({\rm AR}_{n})$ hold with
constants independent of $X$. We exemplify this for $({\rm
FR}_{n})\Rightarrow({\rm SII}_{n})$. If ${\mathscr{X}}$ denotes the class of
all proper metric spaces satisfying $({\rm CI}_{n})[c]$ and $({\rm FR}_{n})$
for some fixed $c>0$ and $h=h(C)$, then Proposition 7.5 shows that every
sequence $(X_{i})_{i\in{\mathbb{N}}}$ in ${\mathscr{X}}$ has asymptotic rank
$\leq n$. Hence, by Theorem 5.4, $({\rm SII}_{n})$ holds for some constant
$M_{0}=M_{0}({\mathscr{X}},n,c,\epsilon)>0$, which depends only on
$n,c,\epsilon$ and the function $h=h(C)$.

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