Coarse Geometry and its Invariants

• Coarse geometry is the study of spaces at large scale, emphasizing uniformly bounded subsets and controlled mappings over fine details.
• The coarse invertibility spectrum is a key invariant that identifies when power maps are coarse equivalences, helping distinguish coarse groups.
• Operator algebras and coarse compactifications extend coarse methods into analysis and topology, providing rigidity and classification tools.

Searching arXiv for recent and relevant papers on coarse geometry and coarse groups. I’ll look up recent arXiv entries on coarse geometry, coarse groups, and related invariants to ground the article. Coarse geometry is the study of spaces at large scale. In this setting, one suppresses small-scale detail and records instead which subsets are uniformly bounded, which relations are controlled, and which maps preserve large-scale structure. The basic objects are coarse spaces, but the subject also includes coarse groups, coarse invariants, coarse compactifications, coarse (co)homology, and operator-algebraic constructions such as Roe algebras. A recent algebraic development is the coarse invertibility spectrum for coarse groups, which records those exponents $n$ for which the power map $g\mapsto g^n$ is a coarse equivalence (Schäfer et al., 2024).

1. Coarse spaces and large-scale equivalence

A coarse structure on a set $X$ is a family $\mathcal E\subseteq \mathcal P(X\times X)$ such that each $E\in\mathcal E$ contains the diagonal $\Delta_X$, is closed under inverse and composition, and is upward closed under inclusion. The elements of $\mathcal E$ are entourages. For $x\in X$, the ball of radius $E$ is $B(x,E)=\{y\in X:(x,y)\in E\}$. A subset is bounded if it is contained in such a ball, and large if some entourage thickening of it covers the whole space (Protasov, 2018).

Maps between coarse spaces are organized by large-scale control rather than pointwise continuity. A map is coarse, or bornologous, if entourages are sent to entourages in the appropriate sense; a bijection whose inverse is also coarse is an asymorphism; and two spaces are coarsely equivalent if they have large subspaces that are asymorphic. These notions provide the basic equivalence relation of the subject (Protasov, 2018).

The category of coarse spaces admits a rigid closure theory. A class $g\mapsto g^n$0 of coarse spaces is a variety if it is closed under subspaces, coarse images, and products. The classification theorem states that every nontrivial proper variety is of the form $g\mapsto g^n$1, the class of all $g\mapsto g^n$2-bounded coarse spaces for some infinite cardinal $g\mapsto g^n$3. In particular, if a variety contains an unbounded metric space, then it is the variety of all coarse spaces (Protasov, 2018).

2. Coarse structures on groups and their generalizations

For groups, coarse geometry interacts directly with algebra. One approach studies compatible coarse structures: a coarse structure $g\mapsto g^n$4 on a group $g\mapsto g^n$5 is compatible if every entourage is contained in a $g\mapsto g^n$6-invariant entourage. Such structures are encoded by generating families $g\mapsto g^n$7 of subsets of $g\mapsto g^n$8, closed under finite unions, products, and inverses. In the Hausdorff topological setting, the family of compact subsets $g\mapsto g^n$9 yields the group-compact coarse structure, a topological analogue of the finitary coarse structure on discrete groups (Nicas et al., 2012).

A second approach uses group ideals. A group ideal $X$0 is an ideal closed under products and inverses, and every left group coarse structure arises uniquely from such an ideal. This formulation makes algebraic control explicit and supports constructions such as $X$1-group coarse structures and coarse structures induced by cardinal invariants. In this setting, quasi-homomorphisms are introduced as a large-scale analogue of homomorphisms: a map $X$2 is a quasi-homomorphism if $X$3 and $X$4 stay uniformly close (Dikranjan et al., 2019).

The notion of coarse group used in the coarse invertibility spectrum is more categorical. There a coarse group is a group object in the coarse category: a coarse space $X$5 equipped with a multiplication $X$6 that is a coarse map, with the group axioms holding up to bounded error. This framework is restrictive: for a metric $X$7 on a group $X$8, multiplication is coarse only when $X$9 is coarsely equivalent to a bi-invariant metric (Schäfer et al., 2024).

The formalism also admits asymmetric generalizations. Semi-coarse structures retain symmetry but not composition closure; quasi-coarse structures retain composition closure but not symmetry; and ordinary coarse structures are exactly those having both properties. This places coarse spaces in a broader hierarchy parallel to semi-uniform and quasi-uniform spaces (Zava, 2018).

3. Invariants in coarse geometry

A central invariant for coarse groups is the spectrum of power invertibility. For a coarse group, the $\mathcal E\subseteq \mathcal P(X\times X)$0-th power map is defined coarsely by iterated multiplication,

$\mathcal E\subseteq \mathcal P(X\times X)$1

Because multiplication is only coarsely associative, $\mathcal E\subseteq \mathcal P(X\times X)$2 is well defined only up to bounded error, but that suffices in the coarse category. The associated invariant is the set of $\mathcal E\subseteq \mathcal P(X\times X)$3 such that $\mathcal E\subseteq \mathcal P(X\times X)$4 is a coarse equivalence. It is preserved by coarse isomorphism, is closed under multiplication and division, and is therefore generated by the primes $\mathcal E\subseteq \mathcal P(X\times X)$5 for which $\mathcal E\subseteq \mathcal P(X\times X)$6 is a coarse equivalence. The corresponding prime spectrum records precisely those coarse symmetries arising from taking prime powers (Schäfer et al., 2024).

Other coarse invariants capture dynamical and dimensional information. Coarse entropy is defined by counting $\mathcal E\subseteq \mathcal P(X\times X)$7-separated families of $\mathcal E\subseteq \mathcal P(X\times X)$8-pseudoorbits and passing to an asymptotic growth rate. It is invariant under coarse equivalence and, for every metric space, can only be either $\mathcal E\subseteq \mathcal P(X\times X)$9 or $E\in\mathcal E$0 (Geller et al., 2021). Coarse asymptotic property $E\in\mathcal E$1 and decomposition complexity translate metric large-scale finiteness conditions into entourage language; in particular, coarse property $E\in\mathcal E$2 implies coarse property $E\in\mathcal E$3 (Bell et al., 2016).

Coarse homology and cohomology provide another invariant layer. For coarse $E\in\mathcal E$4 spaces, coarse Alexander duality takes the form

$E\in\mathcal E$5

and this yields dimensional obstructions to coarse separation. If $E\in\mathcal E$6 coarsely separates a coarse $E\in\mathcal E$7 space $E\in\mathcal E$8, then $E\in\mathcal E$9 (Banerjee et al., 2023, Patil, 26 Jun 2025).

4. The coarse invertibility spectrum on $\Delta_X$0

The first major application of the invertibility spectrum concerns word metrics on $\Delta_X$1 defined by the infinite generating sets

$\Delta_X$2

For the resulting coarse group $\Delta_X$3, the prime spectrum is exactly the set of prime divisors of $\Delta_X$4: $\Delta_X$5 Two structural facts drive the proof. If $\Delta_X$6, then $\Delta_X$7 is not coarsely invertible. If $\Delta_X$8 itself is used as the exponent, then $\Delta_X$9 is coarsely invertible, with coarse inverse

$\mathcal E$0

It follows that if $\mathcal E$1 and $\mathcal E$2 are coarsely isomorphic, then $\mathcal E$3 and $\mathcal E$4 have the same prime divisors. In particular, $\mathcal E$5 and $\mathcal E$6 are not coarsely isomorphic (Schäfer et al., 2024).

A second family comes from profinite topologies. For a nonempty set of primes $\mathcal E$7, the pro-$\mathcal E$8 completion

$\mathcal E$9

induces a coarse structure $x\in X$0 on $x\in X$1. In this case the prime spectrum recovers the defining prime set exactly: $x\in X$2 If $x\in X$3, multiplication by $x\in X$4 is not proper for the $x\in X$5-adic topology and hence is not a coarse equivalence. If $x\in X$6, the map

$x\in X$7

is a coarse inverse to $x\in X$8. Consequently, $x\in X$9 as coarse groups whenever $E$0. These results answer two questions posed in the literature and show that the invertibility spectrum detects the prime data defining both classes of examples (Schäfer et al., 2024).

A remaining classification problem is explicitly left open: whether $E$1 and $E$2 are coarsely isomorphic if and only if $E$3 for some $E$4 (Schäfer et al., 2024).

5. Complements, compactifications, and proximity

Coarse geometry also studies what remains after removing a subset. The coarse cohomology of the complement of $E$5 is defined using expanding neighborhoods

$E$6

where $E$7 is proper. This yields coarse chain and cochain complexes for $E$8, together with a model space $E$9 defined from the pseudometric

$B(x,E)=\{y\in X:(x,y)\in E\}$0

The quotient map $B(x,E)=\{y\in X:(x,y)\in E\}$1 is a coarse equivalence, and the coarse (co)homology of $B(x,E)=\{y\in X:(x,y)\in E\}$2 agrees with the coarse (co)homology of the complement (Banerjee et al., 2023).

An equivariant refinement applies to configuration spaces. For the diagonal $B(x,E)=\{y\in X:(x,y)\in E\}$3 and the $B(x,E)=\{y\in X:(x,y)\in E\}$4-action swapping coordinates, the coarse cohomology of configuration space is defined by

$B(x,E)=\{y\in X:(x,y)\in E\}$5

The resulting coarse van Kampen obstruction yields the coarse obstruction dimension $B(x,E)=\{y\in X:(x,y)\in E\}$6, and if $B(x,E)=\{y\in X:(x,y)\in E\}$7 admits a coarse embedding into $B(x,E)=\{y\in X:(x,y)\in E\}$8, then $B(x,E)=\{y\in X:(x,y)\in E\}$9 (Banerjee, 2024).

At the boundary of a proper metric space, coarse compactifications admit two equivalent descriptions. A compactification is coarse precisely when its extendable bounded continuous functions are Higson, and equivalently when the induced relation on subsets is a large-scale proximity relation. The Higson compactification is universal among coarse compactifications, while the Freudenthal compactification is universal among those with totally disconnected boundary (Hartmann, 2020). The corresponding proximity theory has an intrinsic coarse-space formulation: a connected coarse space induces a coarse proximity if and only if it is coarsely normal, a condition equivalent to earlier notions of large scale normality and asymptotic normality (Grzegrzolka et al., 2018).

6. Operator algebras, descent, and rigidity

Roe-algebraic methods translate coarse geometry into operator theory. For general coarse spaces, Roe algebras can be constructed using coarse geometric modules and coarse supports of operators, extending the classical metric-space theory and covering both uniform and non-uniform Roe algebras, controlled-propagation algebras, and quasi-local operator algebras. The assignment of $g\mapsto g^n$00-theory groups of Roe algebras is a natural functorial operation on coarse spaces (Martínez et al., 2023).

This operator-algebraic perspective yields rigidity results. For coarse disjoint unions of expander graphs, a coarse equivalence is close to a bijective coarse equivalence exactly when, after discarding finitely many components, it matches components by cardinality and sends each such component onto a corresponding component. As an application, if the uniform Roe algebras of coarse disjoint unions of expander graphs are isomorphic, then the underlying spaces are bijectively coarsely equivalent (Baudier et al., 2023).

A broader homotopy-theoretic framework is provided by coarsely excisive functors. Such a functor on coarse spaces yields a coarse assembly map

$g\mapsto g^n$01

and the coarse isomorphism conjecture asserts that $g\mapsto g^n$02 is a stable equivalence for uniformly contractible spaces of bounded geometry. Under appropriate equivariant hypotheses, this implies injectivity of the associated equivariant assembly maps, including instances of the Farrell–Jones and Baum–Connes assembly maps (Mitchener, 2010).

A plausible implication of these developments is that modern coarse geometry is no longer only a theory of quasi-isometry classes of metric spaces. It has become a multi-layered framework in which algebraic self-maps, asymptotic dimension, coarse (co)homology, boundaries, and operator algebras all serve as large-scale invariants. Within that landscape, the coarse invertibility spectrum stands out as an especially elementary but effective invariant: in the examples of $g\mapsto g^n$03 studied so far, prime-level data alone already distinguishes coarse groups that are difficult to separate by more geometric means (Schäfer et al., 2024).