Roller–Fioravanti Compactification

• Roller–Fioravanti Compactification is a canonical compactification of median algebras and spaces, preserving convexity and median structure through compact intervals.
• It employs two equivalent constructions—ultrafilter (Roller) and interval-product (Fioravanti)—to achieve universal compact median completion.
• The framework enables functorial extensions of group actions and ensures dynamical tameness, benefiting research in median geometry and topological dynamics.

The Roller–Fioravanti compactification is a canonical compactification of median algebras and median metric spaces with compact intervals. It generalizes the classical Roller boundary for CAT(0) cube complexes to arbitrary (possibly infinite-rank) median spaces, preserving convexity and the median structure. There are two equivalent constructions: one via embedding into a product of intervals using interval retractions, and one via ultrafilters on the poset of halfspaces. This compactification plays a central role in the topological and dynamical study of median spaces and their group actions, in particular providing a functorial and universal compact median completion, and establishing tameness properties for group actions on finite-rank median algebras (Megrelishvili, 4 Jan 2026, Fioravanti, 2017).

1. Median Algebras and Topological Structure

A median algebra is a set $X$ with a ternary operation $m:X^3 \to X$ satisfying symmetry, idempotence, and a form of associativity:

• Symmetry: $m(a,b,c)=m(a,c,b)$, etc.
• Idempotence: $m(a,b,b)=b$.
• Associativity: $m(m(a,b,c),u,v)=m(a,m(b,u,v),m(c,u,v))$.

A topological median algebra $(X,m)$ is equipped with a Hausdorff topology making $m$ continuous. Intervals $[x,y]=\{z \mid m(x,y,z)=z\}$ are convex subsets, and compactness of these intervals is fundamental in the theory. The rank of $X$ is the maximal $n$ such that the Boolean cube $\{0,1\}^n$ embeds as a median subalgebra, equivalently the maximal cardinality of a set of pairwise-crossing walls.

2. Halfspaces, Walls, and Ultrafiltration

A wall is a pair $\{H,H^c\}$ of nonempty convex subsets whose union is $X$; each $H$ is a halfspace. Halfspaces separate points: for any pair of disjoint convex sets there exists a wall separating them. The set of halfspaces $\mathcal{H}(X)$ carries the structure of a poset with involution $H \mapsto H^c$ and a separation metric $d_\mathcal{H}(x,y)=|\mathcal{H}(x|y)|$, where $\mathcal{H}(x|y)$ is the set of halfspaces separating $x$ from $y$ (Fioravanti, 2017).

Ultrafilters on $\mathcal{H}(X)$ play a central role in the boundary theory. Each $x \in X$ defines a principal ultrafilter $\omega_x = \{H \mid x \in H\}$; non-principal ultrafilters correspond to boundary points in the compactification.

3. Two Equivalent Constructions

• Ultrafilter (Roller) Model The diagonal embedding $\iota:X \to \{0,1\}^{\mathcal{H}(X)}$, $\iota(x)=(\chi_H(x))_{H \in \mathcal{H}(X)}$ sends each $x$ to its characteristic function on halfspaces. The closure $\overline{\iota(X)}$ naturally identifies with the set of ultrafilters on $\mathcal{H}(X)$; $X$ embeds as the set of principal ultrafilters. Median extends coordinatewise, yielding a compact, locally convex, topological median algebra.
• Interval-Product (Fioravanti) Model The family of retractions $\phi_{a,b}: X \to [a,b]$, $\phi_{a,b}(x)=m(a,b,x)$ are continuous and median-preserving. The embedding $\nu:X \to \prod_{(a,b) \in X^2} [a,b]$, $\nu(x) = (m(a,b,x))_{(a,b)}$ lands in a compact product (by Tychonoff). The closure $\overline{\nu(X)}$ forms the Roller–Fioravanti compactification, retaining the median structure and convexity.

Both constructions are equivalent in the presence of compact intervals (Megrelishvili, 4 Jan 2026, Fioravanti, 2017).

4. Universal Properties and Functoriality

The Roller–Fioravanti compactification $\overline{X}^{RF}$ is characterized by two universal properties:

• Universal Compact Median Completion:

For any continuous median-homomorphism $f:X \to Y$ to a compact, locally convex, topological median algebra $Y$, there is a unique continuous extension $\overline{f}:\overline{X}^{RF} \to Y$ which is median-preserving (Fioravanti, 2017).

• Functoriality:

Any median homomorphism $h:X \to Y$ between median algebras with compact intervals lifts to a continuous median homomorphism $$\overline{h}:\overline{X}^{RF} \to \overline{Y}^{RF}$$. Automorphisms extend uniquely; group actions can be extended functorially.

A plausible implication is that this functorial property allows transfer of dynamical and convexity-theoretic results from $X$ to its compactification without loss of structure.

5. Dynamical Tameness and Group Actions

Finite-rank compact median algebras admit important dynamical properties under group actions:

• Rosenthal Representability and Tameness:

For a finite-rank topological median $G$-algebra with compact intervals, the natural action of $G$ by median automorphisms on the Roller–Fioravanti compactification is Rosenthal representable; if the compactification is metrizable, the $G$-system is tame (Megrelishvili, 4 Jan 2026).

• Extension of Group Actions:

Any continuous action of $G$ by median automorphisms on $X$ extends to a dynamically tame action on $\overline{X}^{RF}$, preserving median retractions and compactness of intervals.

This suggests that the Roller–Fioravanti compactification provides a canonical setting for studying tame dynamics of groups acting on median spaces.

6. Compactness, Convergence, and Boundary Behavior

$\overline{X}^{RF}$ is compact by Tychonoff’s theorem and all halfspaces are closed (gate-convex). The original space $X$ embeds continuously, convexly, and densely. Points of the boundary correspond to non-principal ultrafilters; convergence in the compactification is characterized by stabilization of halfspace membership:

$$|\omega \triangle \omega_\alpha| \to 0 \quad \Leftrightarrow \quad \forall h \in \mathcal{H},\; \omega_\alpha(h) = \omega(h) \text{ eventually}.$$

Halfspace-distance $$D(\omega_1,\omega_2) = \frac{1}{2}|\omega_1 \triangle \omega_2|$$ generalizes the metric structure from $X$ to its compactification.

7. Relationship to Roller Boundaries and CAT(0) Cube Complexes

In the case where $X$ is the $0$-skeleton of a CAT(0) cube complex,

• The set of halfspaces $\mathcal{H}$ is discrete and countable.
• The ultrafilter construction recovers Roller’s classical boundary (Fioravanti, 2017).
• Boundary points correspond to ultrafilters with infinite descending chains of halfspaces.
• The coordinatewise convergence matches the standard topology of CAT(0) cube complex boundaries.

This demonstrates that the Roller–Fioravanti compactification unifies boundary constructions in median geometry, generalizing crucial results from cube complexes to the broader class of median algebras and spaces with compact intervals.

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Summary Table: Constructions of Roller–Fioravanti Compactification

Construction Model	Core Objects	Boundary Points
Ultrafilter (Roller)	$\{0,1\}^{\mathcal{H}(X)}$	Non-principal ultrafilters
Interval-Product (Fioravanti)	$\prod_{(a,b)\in X^2}[a,b]$	Non-principal limits in product
CAT(0) cube complex	Cubical ultrafilter structure	Infinite chains of halfspaces

The equivalence of these constructions supplies both a combinatorial (ultrafilter-based) and geometric (interval-based) perspective on median compactifications. The Roller–Fioravanti compactification thus provides foundational infrastructure for modern research in median geometry, convexity, topological dynamics, and the analysis of metrizable group actions (Megrelishvili, 4 Jan 2026, Fioravanti, 2017).