Geometries of topological groups

Abstract

The paper provides an overarching framework for the study of some of the intrinsic geometries that a topological group may carry. An initial analysis is based on geometric nonlinear functional analysis, that is, the study of Banach spaces as metric spaces up to various notions of isomorphism, such as bi-Lipschitz equivalence, uniform homeomorphism, and coarse equivalence. This motivates the introduction of the various geometric categories applicable to all topological groups, namely, their uniform and coarse structure, along with those applicable to a more select class, that is, (local) Lipschitz and quasimetric structure. Our study touches on Lie theory, geometric group theory, and geometric nonlinear functional analysis and makes evident that these can all be seen as instances of a single coherent theory.

The aim of the present paper is to organise and put into a coherent form a number of old and new results, ideas and research programmes regarding topological groups and their linear counterparts, namely Banach spaces. As the title indicates, our focus will be on geometries by which we understand the various types of geometric structures that a Banach space or a topological group may be equipped with, e.g., Lipschitz structure or the quasimetric structure underlying geometric group theory. We shall attempt to provide a common framework and language for several different currently very active disciplines, including geometric nonlinear functional analysis and geometric group theory, and varied objects, e.g., Banach spaces, finitely generated, Lie, totally disconnected locally compact, and Polish groups. For this reason, it will be useful initially not to restrict the objects we consider.

1. Banach spaces as geometric objects

1.1. Categories of geometric structures

Our model example of topological groups, namely, the additive topological group $(X,+)$ underlying a Banach space $(X,\lVert \cdot \rVert )$ is perhaps somewhat unconventional. Certainly, the Banach space $(X,\lVert \cdot \rVert )$ is far more structured than $(X,+)$ and thus one misses much important information by leaving out the normed linear structure. Moreover, algebraically $(X,+)$ is just too simple to be of much interest. However, Banach spaces are good examples since they are objects that have classically been studied under a variety of different perspectives, e.g., as topological vector spaces, as metric or as uniform spaces. So, apart from their intrinsic interest, Banach spaces will illustrate some of the appropriate categories in which to study topological groups and also will provide a valuable lesson in how rigidity results allow us to reconstruct forgotten structure.

The language of category theory will be convenient to formulate the various geometric structures we shall be studying. Recall that to define a category, we need to specify the objects and the morphisms between them. In that way, we derive the concept of isomorphism. Namely, an isomorphism between objects $X$ and $Y$ is a morphism $X\xrightarrow {\phi } Y$ so that, for some morphism $Y\xrightarrow {\psi } X$, both $\psi \phi$ and $\phi \psi$ equal the unique identities on $X$ and $Y$, respectively.

On the other hand, embedding, i.e., isomorphism with a substructure, is not readily a categorical notion as it relies on the model theoretical concept of substructure. However, in all our examples, what constitutes a substructure is evident, e.g., a substructure of a topological vector space is a linear subspace with the induced topology, while a substructure of a metric space is just a subset with the restricted metric. So, for example, an embedding of topological vector spaces is linear map $X\xrightarrow {T} Y$, which is a homeomorphism with its image $T[X]\subseteq Y$.

1.2. Metric spaces

Recall that a Banach space is a complete normed vector space $(X,\lVert \cdot \rVert )$. Thus, the norm is part of the given data. For simplicity, all Banach spaces are assumed to be real, i.e., over the field $\mathbb{R}$. In the strictest sense, an isomorphism should be a surjective linear isometry between Banach spaces, and the proper notion of morphism is thus linear isometry, i.e., a linear operator $X\xrightarrow {T} Y$ so that $\lVert Tx\rVert =\lVert x\rVert$.

However, instead of normed vector spaces, quite often Banach spaces are considered in the weaker category of topological vector spaces with morphisms simply being continuous linear operators. The procedure of dropping the norm from a normed linear space while retaining the topology thus amounts to a forgetful functor

$$\begin{equation*} {\mathsf{NVS}}\xrightarrow {\mathbf{F}} {\mathsf{T}VS} \end{equation*}$$

from the category of normed vector spaces to the category of topological vector spaces. Similarly, rather than entirely eliminating the norm, we may instead erase the linear structure while recording the induced norm metric and thus obtain a forgetful functor

$$\begin{equation*} {\mathsf{NVS}}\xrightarrow {\mathbf{G}} \text{\textsf{Metric Spaces}} \end{equation*}$$

to the category of metric spaces whose morphisms are (not necessarily surjective) isometries. Observe also that these functors preserve embeddings.

This latter erasure however points to our first rigidity phenomenon, namely, the Mazur–Ulam theorem. Indeed, S. Mazur and S. Ulam 39 showed that, if $X\xrightarrow {\phi } Y$ is a surjective isometry between Banach spaces, then $\phi$ is necessarily affine, i.e., the map $Tx=\phi ( x)-\phi (0)$ is a surjective linear isometry between $X$ and $Y$. In particular, any two isometric Banach spaces are automatically linearly isometric.

In a more recent breakthrough 22, G. Godefroy and N. J. Kalton established a similar rigidity result for separable Banach spaces.

Theorem 1.1 (22, Corollary 3.3).

If $X\xrightarrow {\phi } Y$ is an isometric embedding from a separable Banach space $X$ into a Banach space $Y$, then there is an isometric linear embedding of $X$ into $Y$.

Observe that the conclusion here is somewhat weaker than in the Mazur–Ulam theorem, since $\phi$ itself may not be affine. This is for good reasons as, for example, the map $\phi (x)=(x, \sin x)$ is an isometric, but clearly nonaffine embedding of $\mathbb{R}$ into $\ell ^\infty (2)=(\mathbb{R}^2, \lVert \cdot \rVert _\infty )$. Also, the assumption that $X$ is separable is known to be necessary as there are counterexamples in the nonseparable setting (see 22, Corollary 4.4).

Although these two rigidity results do not provide us with a functor from the category of metric space reducts of separable Banach spaces to the category of normed vector spaces, they do show that an isomorphism or embedding in the weaker category of metric spaces implies the existence of an isomorphism, respectively, embedding in the category of normed vector spaces.

1.3. Lipschitz structures

To venture beyond these simple examples, we consider some common types of maps between metric spaces.

Definition 1.2.

A map $X\xrightarrow {\phi } M$ between metric spaces $(X,d)$ and $(M,\partial )$ is

•

Lipschitz if there is a constant $K$ so that, for all $x,y\in X$,$$\begin{equation*} \partial (\phi x, \phi y)\leqslant K\cdot d(x,y); \end{equation*}$$

•

Lipschitz for large distances if there is a constant $K$ so that, for all $x,y\in X$,$$\begin{equation*} \partial (\phi x, \phi y)\leqslant K\cdot d(x,y) +K; \end{equation*}$$

•

Lipschitz for short distances if there are constants $K,\delta >0$ so that$$\begin{equation*} \partial (\phi x, \phi y)\leqslant K\cdot d(x,y) \end{equation*}$$

whenever $x,y\in X$ satisfy $d(x,y)\leqslant \delta$.

A fact that will become important later on is that our definitions above provide a splitting of being Lipschitz as the conjunction of two weaker conditions. Namely, we have the following simple fact:

$$\begin{equation*} \phi \text{ is Lipschitz }\Leftrightarrow \phi \text{ is Lipschitz for both large and short distances.} \end{equation*}$$

As the composition of two Lipschitz maps is again Lipschitz, the class of metric spaces also forms a category where the morphisms are now Lipschitz maps. Similarly with Lipschitz for both large and short distances. However, for later purposes where there are no canonical metrics, it is better not to treat spaces with specific choices of metrics, but rather equivalence classes of these. We therefore define three equivalence relations, namely, bi-Lipschitz, quasi-isometric, and local bi-Lipschitz equivalence on the collection of all metrics on a set $X$ by letting

$$\begin{align*} d\sim _{\mathsf{Lip}} \partial &\Leftrightarrow \big (X,d\big )\underset{\mathsf{id}}{\overset{\mathsf{id}}{\rightleftarrows }} \big (X,\partial \big )\text{ are both Lipschitz}\\ &\Leftrightarrow \exists K\;\; \frac{1}{K} d\leqslant \partial \leqslant K \cdot d,\\ d\sim _{\mathsf{QI}} \partial &\Leftrightarrow \big (X,d\big )\underset{\mathsf{id}}{\overset{\mathsf{id}}{\rightleftarrows }} \big (X,\partial \big )\text{ are Lipschitz for large distances}\\ &\Leftrightarrow \exists K\;\; \frac{1}{K} d- K\leqslant \partial \leqslant K \cdot d +K,\\ d\sim _{\mathsf{locLip}} \partial &\Leftrightarrow \big (X,d\big )\underset{\mathsf{id}}{\overset{\mathsf{id}}{\rightleftarrows }} \big (X,\partial \big )\text{ are Lipschitz for short distances}. \end{align*}$$

Example 1.3.

The standard euclidean metric $d_1(x,y)=|x-y|$ on $\mathbb{R}$ is locally bi-Lipschitz equivalent with the truncated metric $d_2(x,y)=\min \{1,|x-y|\}$. On the other hand, since the map $x\mapsto \sqrt x$ is not Lipschitz for short distances, these are not locally bi-Lipschitz equivalent with the metric

$$\begin{equation*} d_3(x,y)=\sqrt {|x-y|}. \end{equation*}$$

Eventually, when we turn to topological groups, we may occasionally pick out equivalence classes of metrics without being able to choose any particular metric. These thus become objects of the following types.

Definition 1.4.

A Lipschitz, quasimetric, respectively local Lipschitz space is a set $X$ equipped with a bi-Lipschitz, quasi-isometric, respectively local bi-Lipschitz equivalence class $\mathcal{D}$ of metrics on $X$.

In none of these three cases do we have an easy grasp of what the space actually is. By definition, it is that which is invariant under a certain class of transformations. On the other hand, morphisms are simpler. Indeed, a morphism

$$\begin{equation*} \big (X,\mathcal{D}_X\big ) \xrightarrow {\phi } \big (M,\mathcal{D}_M\big ) \end{equation*}$$

between two Lipschitz or local Lipschitz spaces is a map $X\xrightarrow {\phi } M$ that is Lipschitz, respectively Lipschitz for short distances, with respect to some or equivalently any choice of metrics from the respective equivalence classes $\mathcal{D}_X$ and $\mathcal{D}_M$. In this way, Lipschitz and local Lipschitz spaces form categories in which the isomorphisms are bijective functions that are Lipschitz (for short distances) with an inverse that is also Lipschitz (for short distances).

Just as maps that are Lipschitz for large distances need not be continuous and hence fail to capture topological notions, isomorphisms between quasimetric spaces should neither preserve topology nor record spaces’ cardinality either. In analogy with homotopy equivalence of topological spaces, we therefore adjust the notion of morphism.

Definition 1.5.

Two maps $X\xrightarrow {\phi , \psi } M$ from a set $X$ to a metric space $(M,d)$ are close if

$$\begin{equation*} \sup _{x\in X}d\big (\phi x, \psi x\big )<\infty . \end{equation*}$$

Observe that whether $\phi$ and $\psi$ are close depends only on the quasi-isometry class of the metric $d$ on $M$. We may therefore define morphisms in the category of quasimetric spaces to be closeness classes of Lipschitz for large distances maps between these spaces and where composition is computed by composing representatives of these classes.

As a consequence, a Lipschitz for large distances map $X\xrightarrow {\phi } M$ between two quasimetric spaces is a closeness representative of an isomorphism between $X$ and $M$ exactly when there is $M\xrightarrow {\psi } X$, Lipschitz for large distances, so that both $\psi \phi$ and $\phi \psi$ are close to the identities on $X$ and $M$, respectively, i.e., so that

$$\begin{equation*} \sup _{x\in X}d\big (\psi \phi (x), x\big )<\infty \quad \text{ and }\quad \sup _{z\in M}\partial \big (\phi \psi (z), z\big )<\infty \end{equation*}$$

for some/any choice of compatible metrics $d, \partial$ on $X$ and $M$.

Whereas motivating the discussion of isomorphisms here, in practice we shall often avoid equivalence classes of metrics and maps and simply work with representatives from these classes. In this way, a map between metric spaces is called a quasi-isometry if it is a representative for an isomorphism between the associated quasimetric spaces.

Example 1.6.

The map $\mathbb{R}^n\xrightarrow {\phi } \mathbb{Z}^n$ given by $\phi (x_1,\ldots , x_n)=\big (\lfloor x_1\rfloor , \ldots , \lfloor x_n\rfloor \big )$ is a quasi-isometry whose inverse is the inclusion map $\mathbb{Z}^n\to \mathbb{R}^n$ when both are given the euclidean metric.

It is obvious that every metric $d$ on a set $X$ induces not only a metric space $(X,d)$ but also a Lipschitz, locally Lipschitz, and quasimetric space by taking the respective equivalence classes of the metric. Moreover, because the morphisms in the category of a metric space are (not necessarily surjective) isometries, these are also automatically morphisms in the other categories.

On the other hand, whereas not every topological vector space $X$ has a Lipschitz structure compatible with its topology, if $X$ happens to be the reduct of a normed vector space, then all norms compatible with the topology on $X$ are bi-Lipschitz equivalent and thus $X$ is naturally equipped with the Lipschitz structure induced by these norms. This is just a consequence of the simple fact that a continuous linear operator between normed spaces is bounded and therefore Lipschitz.

Although there are counter-examples in the nonseparable case (see 9, Example 7.12), the outstanding problem regarding Lipschitz structure on Banach spaces is whether this completely determines the linear structure.

Problem 1.7.

Suppose $X$ and $Y$ are bi-Lipschitz equivalent separable Banach spaces. Must $X$ and $Y$ also be isomorphic as topological vector spaces?

Even though it is generally felt that the answer should be negative, there are several partial positive results, e.g., 26 and 23. Foremost among these is the following.

Theorem 1.8 (S. Heinrich and P. Mankiewicz 26, Theorem 2.6).

Suppose $X$ and $Y$ are bi-Lipschitz equivalent separable dual Banach spaces and assume that $X\cong X \oplus X$ and $Y\cong Y\oplus Y$ as topological vector spaces. Then $X$ and $Y$ are isomorphic as topological vector spaces.

This applies, for example, to reflexive spaces such as $\ell ^p$ and $L^p([0,1])$ for $1<p<\infty$.

1.4. Banach spaces as uniform spaces

Evidently, every map between metric spaces that is Lipschitz for short distances is automatically uniformly continuous. In particular, this means that the uniform structures $\mathcal{U}_d$ and $\mathcal{U}_\partial$ given by two locally Lipschitz equivalent metrics $d$ and $\partial$ must coincide, i.e., $\mathcal{U}_d=\mathcal{U}_\partial$. However, to give a proper presentation of this and also to motivate the category of coarse spaces, recall the definition of uniform structures.

Definition 1.9 (A. Weil 55).

A uniform space is a set $X$ equipped with a filter $\mathcal{U}$ of subsets $E\subseteq X\times X$, called entourages, satisfying

(1)

$\Delta \subseteq E$ for all $E\in \mathcal{U}$;

(2)

if $E\in \mathcal{U}$, then $E^{-1}=\{(y,x)\mid (x,y)\in E\}\in \mathcal{U}$;

(3)

if $E\in \mathcal{U}$, then $F\circ F=\{(x,z)\mid \exists y\, (x,y),(y,z)\in F\}\subseteq E$ for some $F\in \mathcal{U}$.

Here $\Delta =\{(x,x)\mid x\in X\}$ denotes the diagonal in $X\times X$. Recall that if $d$ is an écart (a.k.a. pseudo-, pre-, or semimetric) on a set $X$ (i.e., $d$ is a metric except that possibly $d(x,y)=0$ for distinct $x,y\in X$), then the induced uniform structure $\mathcal{U}_d$ is the filter generated by the family of entourages

$$\begin{equation*} E_\alpha =\{(x,y)\mid d(x,y)<\alpha \} \end{equation*}$$

for $\alpha \; {\color[RGB]{191, 0, 0} >0}$.

Also, a morphism between two uniform spaces $(X,\mathcal{U})$ and $(M,\mathcal{V})$ is simply a uniformly continuous map $X\xrightarrow {\phi } M$, that is, satisfying

$$\begin{equation*} \forall F\in \mathcal{V}\;\ \exists E\in \mathcal{U}\colon (x,y)\in E \Rightarrow (\phi x, \phi y)\in F. \end{equation*}$$

Again, as the notion of substructure is apparent, we obtain a notion of uniform embeddings, namely, isomorphism with a substructure.

Important early work on the uniform classification of Banach spaces was done by P. Enflo, J. Lindenstrauss, and M. Ribe, who established a number of rigidity results for these. For example, the combined results of Lindenstrass 35 and Enflo 14 establish that if $1\leqslant p<q<\infty$, then the spaces $L^p([0,1])$ and $L^q([0,1])$ are not uniformly homeomorphic. However, whereas this distinguishes between the $L^p$ spaces, it does not tell an $L^p$ space apart from an arbitrary space. Regarding this, W. B. Johnson, J. Lindenstrauss, and G. Schechtman 28 show that if a Banach space $X$ is uniformly homeomorphic to $\ell ^p$ for some $1<p<\infty$, then $X$ is actually isomorphic to $\ell ^p$ as topological vector spaces. Considering instead uniform embeddings, let us just mention the result of Enflo 15 stating that not every separable Banach space embeds uniformly into $\ell ^2$.

For the record, let us mention that, as opposed to the Lipschitz category, it is known that the uniform structure does not determine the linear structure even in the separable case. Namely, by work of Ribe 45, there are examples of separable uniformly homeomorphic Banach spaces that are not isomorphic as topological vector spaces. Similarly, quasimetric structure does not determine uniform structure. Indeed by a result due to Kalton 32 there are separable quasi-isometric Banach spaces that are not uniformly homeomorphic.

1.5. Banach spaces as coarse spaces

Although we have not discussed Banach spaces viewed as quasimetric spaces, we shall now consider an even weaker category that abstracts large scale content from metric spaces in a manner similar to how uniform spaces abstract small scale content. In fact, the following definition is an almost perfect large scale counterpart to that of uniform spaces.

Definition 1.10 (J. Roe 46).

A coarse space is a set $X$ equipped with an ideal $\mathcal{E}$ of entourages $E\subseteq X\times X$ satisfying

(1)

$\Delta \in \mathcal{E}$;

(2)

if $E\in \mathcal{E}$, then $E^{-1}\in \mathcal{E}$;

(3)

if $E\in \mathcal{E}$, then $E\circ E\in \mathcal{E}$.

Again, if $(X,d)$ is a pseudometric space, the associated coarse structure $\mathcal{E}_d$ is then the ideal generated by the entourages $E_\alpha =\{(x,y)\in X\times X\mid d(x,y)<\alpha \}$, where now we require $\alpha \; {\color[RGB]{191, 0, 0}<\infty }$ rather than $\alpha >0$.

In particular, this means that we can define two maps $Y\xrightarrow {\phi , \psi } X$ from a set $Y$ into a coarse space $(X,\mathcal{E})$ to be close if there is an entourage $E\in \mathcal{E}$ so that $(\phi y, \psi y)\in E$ for all $y\in Y$. This conservatively extends the definition of closeness from the case of metric spaces.

Definition 1.11.

A map $X\xrightarrow {\phi } M$ between two coarse spaces $(X,\mathcal{E})$ and $(M,\mathcal{F})$ is bornologous if

$$\begin{equation*} \forall E\in \mathcal{E}\;\ \exists F\in \mathcal{F}\colon (x,y)\in E \Rightarrow (\phi x, \phi y)\in F. \end{equation*}$$

It follows that a map $(X,d)\xrightarrow {\phi }(M,\partial )$ between pseudometric spaces is bornologous if and only if there is a monotone increasing function $\omega \colon \mathbb{R}_+\to \mathbb{R}_+$ so that

$$\begin{equation*} \partial (\phi x,\phi y)\leqslant \omega \big (d(x,y)\big ) \end{equation*}$$

for all $x,y\in X$.

Analogously to the category of quasimetric spaces, morphisms between coarse spaces are closeness classes of bornologous maps, and so two coarse spaces $(X,\mathcal{E})$ and $(Y,\mathcal{F})$ are coarsely equivalent (that is, isomorphic as coarse spaces) if there are bornologous maps $X\underset{\phi }{\overset{\psi }{\rightleftarrows }}Y$ so that $\psi \phi$ and $\phi \psi$ are close to the identities on $X$ and $Y$, respectively.

More concretely, note that a map $X\xrightarrow {\phi } M$ from a metric space $(X, d)$ into a metric space $(M, \partial )$ is a uniform embedding if

$$\begin{equation*} d(x_n,y_n)\to 0 \Leftrightarrow \partial (\phi x_n, \phi y_n)\to 0 \end{equation*}$$

for all sequences $x_n, y_n\in X$. In the same manner, $X\xrightarrow {\phi } M$ is a coarse embedding if, for all $x_n,y_n$,

$$\begin{equation*} d(x_n,y_n)\to \infty \Leftrightarrow \partial (\phi x_n, \phi y_n)\to \infty . \end{equation*}$$

A coarse embedding is then a coarse equivalence⁠¹ if furthermore $\phi [X]$ is cobounded in $M$, i.e.,

¹

Strictly speaking, $\phi$ is a closeness representative of a coarse embedding.

✖

$$\begin{equation*} \sup _{a\in M}{\mathsf{dist}}(a, \phi [X])<\infty . \end{equation*}$$

Because Lipschitz for short distances entails uniformly continuous and Lipschitz for large distances entails bornologous, we obtain a diagram of forgetful functors between the categories of metric, Lipschitz, local Lipschitz, uniform, quasimetric, and coarse spaces as in Figure 1.

Example 1.12 (Near isometries).

Consider the category of metric spaces in which morphisms are closeness classes of near isometries, i.e., of maps $(X,d)\xrightarrow {\phi } (Y,\partial )$ so that

$$\begin{equation*} \kappa _\phi =\sup _{x,z\in X}|d(x,z)-\partial (\phi x, \phi z)|<\infty . \end{equation*}$$

Then two spaces are isomorphic provided there are near isometries $X\underset{\phi }{\overset{\psi }{\rightleftarrows }} Y$ so that $\psi \phi$ and $\phi \psi$ are close to the identities on $X$ and $Y$, respectively. Observe that, in this category, it is easy to produce isomorphic spaces that are not isometric and also automorphisms that are not close to any auto-isometries.

We remark that, if $X$ and $Y$ are Banach spaces that are isomorphic in this category, then there is a surjective near isometry $X\xrightarrow {\phi } Y$ so that furthermore $\phi (0)=0$. Furthermore, by results due to J. Gevirtz 19 and P. M. Gruber 25, for any such $\phi$, there is a linear isometry $X\xrightarrow {T} Y$ with

$$\begin{equation*} \sup _x\lVert Tx-\phi x\rVert \leqslant 4\kappa _\phi . \end{equation*}$$

In particular, this shows that any isomorphism is close to a surjective linear isometry and hence that the new notion of isomorphism coincides with linear isometry of spaces.

1.6. Rigidity of morphisms and embeddability

So far we have encountered rigidity results for isomorphisms and individual objects in the various categories. The following simple fact, on the other hand, will establish rigidity of morphisms.

Lemma 1.13 (General Corson–Klee lemma).

Suppose $X\xrightarrow {\phi }E$ is a map between normed vector spaces so that, for some $\delta ,\Delta >0$ and all $x,y\in X$,

$$\begin{equation*} \lVert x-y\rVert <\delta \;\Rightarrow \; \lVert \phi x-\phi y\rVert <\Delta . \end{equation*}$$

Then $\phi$ is Lipschitz for large distances.

Proof.

Given $x,y\in X$, let $n$ be minimal so that $\lVert x-y\rVert <n\cdot \delta$. Then there are $v_0=x$, $v_1$, …, $v_n=y$ so that $\lVert v_i-v_{i+1}\rVert <\delta$ for all $i$. It thus follows that

$$\begin{equation*} \lVert \phi x-\phi y\rVert \leqslant \sum _{i=0}^{n-1}\lVert \phi v_i- \phi v_{i+1}\rVert <n\cdot \Delta . \end{equation*}$$

Therefore, $\lVert \phi x-\phi y\rVert <\frac{\Delta }{\delta }\cdot \lVert x-y\rVert +\Delta$.

■

In particular, both a uniformly continuous and a bornologous map between two Banach spaces is automatically Lipschitz for large distances. Similarly, a uniform homeomorphism or a coarse equivalence between Banach spaces is also a quasi-isometry. On the other hand, since a uniform or coarse subspace of a Banach space need not be the reduct of linear subspace itself, a uniform or coarse embedding between Banach spaces is not in general a quasi-isometric embedding.

Remark 1.14 (Reconstruction functors).

The above comments show that, when we restrict our attention to reducts of Banach or just normed vector spaces, there are reconstruction functors going from the categories of uniform, respectively coarse spaces, to quasimetric spaces. Namely, suppose $\mathcal{U}$ is the uniform structure induced from some normed vector space structure on the set $X$. Then we let ${\mathbf{F}}(X,\mathcal{U})=(X, \mathcal{D})$ be the quasimetric space induced by some or, equivalently, any normed vector space structure on the set $X$ that is compatible with the uniformity $\mathcal{U}$. Indeed, if $(X,+, \lVert \cdot \rVert )$ and $(X,\oplus , \left\Vvert \cdot \right\Vvert )$ are two such normed vector space structures, then

$$\begin{equation*} (X,\lVert \cdot \rVert ) \xrightarrow {\mathsf{id}} (X,\left\Vvert \cdot \right\Vvert ) \end{equation*}$$

is a uniform homeomorphism and thus a quasi-isometric equivalence. It thus follows that the quasi-isometric equivalence classes of the norm metrics actually coincide.

Similarly, every map between Banach spaces that is Lipschitz for short distances is automatically Lipschitz for large distances and hence actually Lipschitz (for all distances). So this provides a functor from the category of Banach spaces viewed as local Lipschitz spaces to the category of Banach spaces viewed as Lipschitz spaces.

At this point, we can refer to Figure 2 for a diagram of categories and the functors relating them. All categories refer exclusively to reducts of separable real Banach spaces and the black arrows to functors. Also, dashed blue arrows refer to a rigidity result for isomorphism. For example, an isomorphism in the category of metric spaces induces another isomorphism in the category of normed vector spaces by the Mazur–Ulam theorem.

Again, whereas a functor maps isomorphisms to isomorphisms, it need not preserve embeddings, since the latter notion is not intrinsic to the category. Thus, although a uniform embedding between Banach spaces is bornologous, it need not be a coarse embedding. Nevertheless, we do have rigidity results for embeddings not stemming from functors. Indeed, for separable Banach spaces, by the Godefroy–Kalton theorem, isometric embeddings give rise to other linear isometric embeddings. This rigidity is indicated by a curly red arrow in Figure 2.

Now, even though by 32 there are separable quasi-isometric Banach spaces that are not uniformly homeomorphic, it is an open problem whether the notions of uniform and coarse embeddability between Banach spaces coincide.⁠²

²

The origins of this problem are not entirely clear, but the need for a better understanding of the connection between these notions was noted by Kalton in 31.

✖

Problem 1.15.

Are the following two conditions equivalent for all (separable) Banach spaces $X$ and $E$?

(1)

$X$ uniformly embeds into $E$.

(2)

$X$ coarsely embeds into $E$.

Observe that this is far from being trivial, since it is easy to produce uniform embeddings that are not coarse embeddings, and vice versa. Also, one cannot hope to replace coarse embeddings by quasi-isometric embeddings, since, for example, $\ell ^1$ embeds into $\ell ^2$ uniformly, but not quasi-isometrically.

Theorem 1.16.

Assume $X$ and $E$ are Banach spaces and that $E\oplus E$ embeds as a topological vector space into $E$. Suppose also $X\xrightarrow {\phi }E$ is uniformly continuous and that, for some $\delta ,\Delta >0$,

$$\begin{equation*} \lVert x-y\rVert >\Delta \;\Rightarrow \; \lVert \phi x-\phi y\rVert >\delta . \end{equation*}$$

Then there is a simultaneously uniform and coarse embedding $X\xrightarrow {\psi }E$.

Proof.

As $E\oplus E$ embeds into $E$, we may inductively construct three sequences $E_n, Z_n, V_n$ of closed linear subspaces of $E$ so that $E_n\cong Z_n \cong E$ as topological vector spaces and

$$\begin{equation*} E_{n+1}\oplus Z_{n+1}\subseteq Z_n \end{equation*}$$

and

$$\begin{equation*} V_n=E_1\oplus E_2\oplus \cdots \oplus E_n\oplus Z_n. \end{equation*}$$

Indeed, we simply begin with an isomorphic copy $V_1$ of $E\oplus E$ inside of $E$, and let $E_1$ and $Z_1$ be the first and second summand, respectively. Again, pick a copy of $E\oplus E$ inside of $Z_1$ with first and second summand denoted respectively $E_2$ and $Z_2$ and let $V_2=E_1\oplus E_2\oplus Z_2\subseteq V_1$, etc.

Let also $P_n$ denote the projection of $V_n$ onto the summand $E_n$ along the decomposition above. While each $P_n$ is bounded, there need not be any uniform bound on their norms. Note now that $V_1\supseteq V_2\supseteq \cdots$, so we can let $V=\bigcap _{n=1}^\infty V_n$, which is a closed linear subspace of $E$ containing all of the $E_n$. Moreover, the $P_n$ all restrict to bounded projections $P_n\colon V\to E_n$ so that $E_m\subseteq \ker P_n$ whenever $n\neq m$.

Composing $\phi$ with linear isomorphisms between $E$ and $E_n$, we get a sequence of uniformly continuous maps $X\xrightarrow {\phi _n} E_n$ satisfying $\lVert x-y\rVert >\Delta _n\;\Rightarrow \; \lVert \phi x-\phi y\rVert >\delta _n$ for some $\Delta _n,\delta _n>0$ and bounded projections $P_n\colon V\to E_n$ so that $E_m\subseteq \ker P_n$ for $n\neq m$. By Lemma 1 of 48, this implies that $X$ admits a simultaneously coarse and uniform embedding into $V$ and thus into $E$.

■

Observe that if $X\xrightarrow {\phi }E$ is either a uniform or coarse embedding between Banach spaces, then there are $\Delta , \delta >0$ as in Theorem 1.16. Therefore, apart from the mild assumption that $E\oplus E$ embeds as a topological vector space into $E$, we have the implication (1)$\Rightarrow$(2) in Problem 1.15.

Corollary 1.17.

Suppose $X$ and $E$ are Banach spaces so that $E\oplus E$ embeds as a topological vector space into $E$. Then, if $X$ uniformly embeds into $E$, $X$ also coarsely embeds into $E$.

On the other hand, if a coarse embedding could always be strengthened to be uniformly continuous, then we would essentially have proved the converse direction (2)$\Rightarrow$(1). However, one must contend with the following serious obstruction.

Theorem 1.18 (A. Naor 42, Theorem 1).

There is a bornologous map $X\xrightarrow {\phi }E$ between separable Banach spaces that is not close to any uniformly continuous map.

The above results indicate that the uniform structure of a Banach space is more rigid than the coarse structure. However, once we pass to the underlying topology, almost no information is left. Indeed, it is a result of M. I. Kadets 29 and H. Toruńczyk 53 that any two infinite-dimensional Banach spaces of the same density character are homeomorphic. Furthermore, in combination with a result of R. D. Anderson 2, it follows that all separable infinite-dimensional Banach spaces are all homeomorphic to the countable product of lines, $\mathbb{R}^\mathbb{N}$.

Remark 1.19 (Universal spaces).

In the various categories above, it is interesting to search for universal spaces, that is, separable spaces into which every other separable spaces embeds. For example, a classical result states that, for $K$ an uncountable compact metric space, $C(K)$ is universal in the category NVS; every separable Banach space admits an isometric linear embedding into $C(K)$. Similarly, by a result of I. Aharoni 1, $c_0$ is universal in the category Lipschitz.

In contradistinction to this, F. Baudier, G. Lancien, and T. Schlumprecht 7 recently showed that there is no infinite-dimensional space that coarsely embeds into all infinite-dimensional spaces. And when combined with a result of Y. Raynaud 43, one sees that the same holds for uniform embeddings.

1.7. Banach spaces as local objects

The results of Enflo, Johnson, Lindenstrauss, and Schechtman 142835 mentioned earlier show rigidity for the uniform structure of the individual spaces $L^p([0,1])$ and $\ell ^p$. However, there is also a beautiful rigidity result due to Ribe encompassing all Banach spaces. To explain this, we need a technical concept.

Definition 1.20.

A Banach space $X$ is said to be crudely finitely representable in a Banach space $Y$ if there is a constant $K$ so that, for every finite-dimensional subspace $E\subseteq X$, there is a finite-dimensional subspace $F\subseteq Y$ and a linear isomorphism $E\xrightarrow {T} F$ with $\lVert T\rVert \cdot \lVert T^{-1}\rVert \leqslant K$.

We then say that $X$ and $Y$ are locally isomorphic in case they are crudely finitely representable in each other. In 44, Ribe then establishes the surprising fact that any two uniformly homeomorphic spaces must be locally isomorphic. In particular, this implies that all local properties of Banach spaces, i.e., that only depend on the finite-dimensional subspaces (up to some uniform constant of isomorphism), are in principle expressible in terms of the uniform structure of the entire space. This, in turn, has motivated to so called Ribe programme (see, e.g., Naor 41) of identifying exclusively metric expressions for these various local invariants of Banach spaces, such as convexity, smoothness, type, and cotype, which furthermore then become applicable not only in the linear setting but to metric spaces in general.

Subsequent proofs of Ribe’s theorem go by showing that if $X$ and $Y$ are quasi-isometric separable spaces, then $X$ and $Y$ have bi-Lipschitz equivalent ultrapowers $X^\mathcal{U}$ and $Y^\mathcal{U}$. Moreover, if $V\xrightarrow {\phi } W$ is a bi-Lipschitz embedding of a separable Banach space $V$ into a Banach space $W$, then, using differentiation techniques, $V$ embeds as a topological vector space into $W^{**}$. In particular, the diagonal copy of $X$ in $X^\mathcal{U}$ embeds as a topological vector space into $(Y^\mathcal{U})^{**}$. Now, by the principle of local reflexivity, $(Y^\mathcal{U})^{**}$ is crudely finitely representable in $Y^\mathcal{U}$ and, by the nature of ultrapowers, $Y^\mathcal{U}$ is crudely finitely representable in $Y$. Combined, this shows that if $X$ and $Y$ are quasi-isometric separable spaces, then $X$ is crudely finitely representable in $Y$ and vice versa, i.e., $X$ and $Y$ are locally isomorphic. As uniformly homeomorphic or coarsely equivalent spaces are also quasi-isometric, Ribe’s theorem follows.

One may think of Banach spaces as objects in the category Local of local spaces in the following sense. The objects of the category are simply separable Banach spaces, and we put an arrow $X\to Y$ from $X$ to $Y$ in case $X$ is crudely finitely representable in $Y$. Observe that, in this way, an arrow $X\to Y$ does not necessarily correspond to the existence of a special type of function from $X$ to $Y$. However, if $X$ isometrically embeds into $Y$, then $X$ also linearly isometrically embeds and thus is crudely finitely representable in $Y$. This means that we obtain a last functor from the category of metric reducts of separable Banach spaces to Local.

In Figure 2, Ribe’s theorem is indicated as an arrow from the category Coarse to Local. His original rigidity theorem (that is, the arrow from Uniform to Local) is then obtained by composition with the functors from Uniform to Quasimetric and further onto Coarse.

When we restrict the category Local to infinite-dimensional spaces, we have initial and terminal objects $X$ and $Y$, that is, so that for every $Z$ there are (trivially unique) arrows

$$\begin{equation*} X\to Z\to Y. \end{equation*}$$

Indeed, by a result of A. Dvoretzky, Hilbert space $\ell ^2$ is crudely finitely representable in every infinite-dimensional Banach space (see 18), whereas, by a result of S. Kwapień 34 any space crudely finitely representable in $\ell ^2$ has type and cotype $2$ and must be isomorphic to $\ell ^2$ as a topological vector space. Thus, up to isomorphism, $\ell ^2$ is the unique initial object.

On the other hand, $Y$ is a terminal object exactly when $\ell ^\infty$ is crudely finitely representable in $Y$, which by a result of B. Maurey and G. Pisier 38 is equivalent to $Y$ only having trivial cotype. This shows that, for example, $c_0$ and the reflexive space

$$\begin{equation*} (\ell ^\infty (2)\oplus \ell ^\infty (3)\oplus \cdots )_{\ell ^2} \end{equation*}$$

are terminal.

For Banach spaces, there are also interesting concepts of minimality of objects, which can be phrased as being an initial object in an appropriate category. Namely, a separable infinite-dimensional Banach space $X$ is said to be minimal if $X$ embeds as a topological vector space into all of its infinite-dimensional closed subspaces. Similarly, $X$ is locally minimal if $X$ is crudely finitely representable in all its closed infinite-dimensional subspaces. Both of these concepts allow for Ramsey style dichotomies that establish canonical obstructions for containing (locally) minimal subspaces (see 17, Theorems 1.1 and 1.2). Specifically, every infinite-dimensional Banach space contains an infinite-dimensional closed linear subspace $X$ satisfying one of the following:

(1)

$X$ is crudely finitely representable in all its infinite-dimensional subspaces;

(2)

$X$ has a Schauder basis $(x_n)_{n=1}^\infty$ so that no infinite-dimensional subspace $Y\subseteq X$ is crudely finitely representable in all tail subspaces $X_m=[x_n]_{n=M}^\infty$ with a uniform constant.

Let us end this section by noting that, particularly through the impetus of J. Bourgain and A. Naor, the nonlinear and metric theory of Banach spaces has blossomed into a very rich theory with deep connections to computer science. An overview of some of these topics can be found in G. Godefroy’s survey 21.

2. Geometric structures on topological groups

2.1. Uniform and local Lipschitz structure

In the preceding section, we have introduced various geometric structures through the instructive example of Banach spaces. In this case, once the categories are understood, there is no discussion of what the appropriate structure of a Banach space is, since it is just obtained by stripping away information. Also, we saw how one may reconstruct, e.g., affine structure from the metric structure and quasimetric structure from the uniform structure. However, for topological groups that do not a priori have this additional structure, the problem is the reverse. Namely, how and when can we endow the abstract topological group with a canonical structure of a given type.

Recall that a topological group is simply a group $G$ equipped with a topology in which the group operations are continuous. Even a Lie group may just be seen as a locally compact, locally euclidean group (in the light of the solution to Hilbert’s fifth problem) and thus simply a special type of topological group without any further differentiable structure. For simplicity, all topological groups will henceforth be assumed to be Hausdorff.

Now, apart from being a topological space, a topological group $G$ also has a couple of canonical uniform structures associated with it. The most interesting in this context in the left-uniform structure $\mathcal{U}_L$, which is the filter on $G\times G$ generated by entourages

$$\begin{equation*} E_V=\{(x,y)\in G\times G\mid x^{-1}y\in V\}, \end{equation*}$$

where $V$ ranges over identity neighbourhoods in $G$. Observe that if $(X,+)$ is the additive topological group of a Banach space, this is simply the uniform structure given by the norm metric.

As always, with uniform spaces it is often useful to work with écarts generating the uniformity and, in the case of groups, one can even require these to be compatible with the algebraic structure. Indeed, an écart $d$ is said to be left-invariant if