Horofunction Compactification

• Horofunction compactification is a method to extend a metric space by introducing a canonical boundary at infinity via asymptotic distance functions.
• It translates analytic and geometric structures into the closed dual unit ball, unifying compactification methods across symmetric spaces, cones, and other settings.
• The construction preserves group actions and elucidates connections between dynamics, duality, and geometric stratification in varied metric contexts.

The horofunction compactification is an analytic and topological procedure that extends a metric space by adding a "boundary at infinity" comprising asymptotic classes of distance functions. For spaces with substantial geometric or algebraic structure—such as noncompact symmetric spaces, normed spaces, symmetric cones, CAT(0) or Gromov hyperbolic spaces, and Teichmüller-type moduli spaces—horofunction compactification offers a canonical and widely applicable concept, with deep connections to duality, dynamics, group actions, and classical compactifications in representation theory and complex analysis.

1. General Definition and Construction

Given a metric space $(X,d)$ and a base point $b\in X$, the horofunction compactification is constructed via the assignment

$$\Phi:X\to \mathrm{Lip}_b(X),\qquad y\mapsto h_y(\cdot) = d(\cdot, y) - d(b, y),$$

where $\mathrm{Lip}_b(X)$ denotes the space of real-valued $1$-Lipschitz functions on $X$ vanishing at $b$, endowed with the topology of pointwise convergence (or uniform convergence on compact sets in the proper case). The closure $$\overline{X} = \overline{\Phi(X)} \subset \mathrm{Lip}_b(X)$$ yields a compactification of $X$, with $\partial X = \overline{X} \setminus \Phi(X)$ termed the horofunction boundary. Under mild properness or separability conditions, every horofunction can be realized as a limit

$h(x) = \lim_{k\to\infty}[d(x, x_k) - d(b, x_k)]$

for some sequence $(x_k)\subset X$ tending to infinity along some almost-geodesic. These boundary points often encode asymptotic geometric or dynamical information (Chu et al., 2022, Lemmens et al., 2021).

2. Horofunction Compactification in Hermitian Symmetric Spaces

Let $M$ be a noncompact Hermitian symmetric space, realized analytically as the open unit ball $D$ of a finite-dimensional JB$^*$-triple $V$ with norm $\|\cdot\|$. The Carathéodory distance $\rho$ on $D$ is a Finsler metric: for $w\in T_v D \cong V$,

$$F_D(v;w) = \lim_{t\to 0^+}\frac{1}{t}\rho(v, v + t w) = \left\|[B(v,v)]^{-1/2} w\right\|,$$

where $B(v,v)$ is the Bergman operator. At the origin $0$, $F_D(0;w) = \|w\|$, so the tangent space inherits the JB$^*$–norm (Chu et al., 2022).

The metric compactification of $(V, \|\cdot\|)$ is naturally identified with the closed dual unit ball $B^*$ of $(V, \|\cdot\|)$. Through the exponential map at the origin,

$$\exp_0: V \to D,\quad \sum_{i=1}^r \lambda_i e_i \mapsto \sum_{i=1}^r \tanh(\lambda_i) e_i,$$

one obtains a homeomorphism between the metric compactifications of $(V,\|\cdot\|)$ and $(D,\rho)$, including the horofunction boundaries. This identification preserves the geometric structure, so that the metric (or horofunction) compactification of $M$ is realized as $B^*$, the closed dual unit ball of the tangent norm (Chu et al., 2022, Lemmens et al., 2021).

3. Explicit Description of Horofunctions and Boundary Stratification

3.1 Flat Model $(V, \|\cdot\|)$

Horofunctions of $(V, \|\cdot\|)$ are given by

$h(x) = \lim_{t\to\infty}[\|x-t v\| - \|-t v\|],$

where $v$ is a direction escaping to infinity. In the spectral decomposition, if $x = \sum_{i=1}^r \lambda_i e_i$, and $e_1, ..., e_r$ are mutually orthogonal minimal tripotents, every horofunction $h \in \partial V$ corresponds to a tripotent $e = \sum_{i=1}^p e_i$ with parameters $a_1 \ge \cdots \ge a_p \ge 0$ and $\min_i a_i = 0$. The horofunction is constructed as

$$h(x) = \max_{\substack{u\in V_2(e)\ \|u\| = 1}} \bigl[-\langle e \circ x + x \circ e, u\rangle - \sum_{i=1}^p a_i \langle e_i \circ e_i, u\rangle\bigr].$$

This stratification corresponds directly to points in the closed dual unit ball, and the boundary is homeomorphic to $B^*$ (Chu et al., 2022).

3.2 Curved Model $(D, \rho)$

For the curved ball $D$, horofunctions arise as limits along sequences in $D$ converging to boundary points. Each horofunction is determined by a tripotent $e = \sum_{i=1}^p e_i \in \partial D$ and parameters $0 < d_1 \le \cdots \le d_p \le 1$ with $\max_i d_i = 1$: $$h(z) = \ln\max_{1 \le i, j \le p}\bigl(d_i d_j \|B(z,z)^{-1/2}B(z,e)P_{ij}\|\bigr),$$ where $P_{ij}$ are the joint Peirce projections (Chu et al., 2022).

4. Dual Ball Correspondence, Exponential Homeomorphism, and Parts

There exists a canonical homeomorphism

$$\Psi: V\cup \partial V \to D^\circ = \{x = \sum_i \mu_i e_i : -1 \le \mu_i \le 1\},$$

with

$$x = \sum_i \lambda_i e_i \mapsto \sum_i \frac{e^{\lambda_i} - e^{-\lambda_i}}{e^{\lambda_i} + e^{-\lambda_i}} e_i,$$

and for a boundary horofunction labeled by $(e, (a_i))$: $h(e, (a_i)) \mapsto \sum_i e^{-a_i} e_i.$ Under the Riesz identification $V^* \cong V$ via the trace form, $D^\circ$ is exactly the closed dual unit ball $B^*$. This correspondence extends to the curved model, via the exponential map at the origin and its boundary extension (Chu et al., 2022).

The stratification of the horofunction boundary into equivalence classes, called "parts," is governed by the facial structure of the dual unit ball. Each part corresponds to the relative interior of a face of $B^*$; the detour metric on the horofunction boundary restricts to faces and controls the geometry of the boundary (Chu et al., 2022, Lemmens et al., 2021).

5. Further Examples: Symmetric Cones, Hilbert Geometry, and Finsler Symmetric Spaces

5.1 Symmetric Cones

For symmetric cones $A_+^\circ$ modeled on Euclidean Jordan algebras, the horofunction compactifications under invariant Finsler metrics such as the Thompson and Hilbert distances have been described explicitly. These compactifications are homeomorphic, under the exponential map, to the horofunction compactification of the tangent space (a JB-algebra), with the boundary parametrized by support data corresponding to faces of the dual ball. In the projective Hilbert case, the horofunction boundary is stratified by faces matching the facial structure of the dual unit ball in the variation norm (Lemmens, 2021).

5.2 Hilbert Geometry

For bounded convex domains $\Omega \subset \mathbb{R}^n$ with the Hilbert metric, the horofunction boundary decomposes into Busemann points corresponding to geodesic directions, with "parts" (equivalence classes of Busemann points at finite detour distance) corresponding to the facial geometry of the dual cone. This structure enables precise identification of isometry groups and aligns the horofunction compactification with the natural geometric and combinatorial compactifications of the domain (Walsh, 2014, Lemmens et al., 2021).

5.3 Symmetric Spaces and Polyhedral Metrics

In noncompact symmetric spaces $G/K$, the horofunction compactification with respect to a $G$-invariant polyhedral Finsler metric has been shown to coincide equivariantly with the generalized Satake compactification for an associated representation. The horofunction boundary is homeomorphic to the dual of the convex polytope governing the unit ball of the Finsler norm in a maximal flat, with the stratification matching Weyl chamber faces (Haettel et al., 2017).

6. Significance and Broader Applications

The horofunction compactification provides a canonical and unifying boundary construction for metric spaces with varied geometric and analytic structures:

• Metric geometry and duality: The homeomorphism with the closed dual unit ball reveals a deep duality between tangent and boundary structures (Chu et al., 2022, Lemmens et al., 2021, Ji et al., 2016).
• Compactification of symmetric and locally symmetric spaces: The approach links algebraic, geometric, and Finsler compactifications under a single metric-analytic framework (Haettel et al., 2017, Lemmens, 2021).
• Comparison with classical boundaries: For Gromov hyperbolic spaces, CAT(0) spaces, and Hermitian symmetric spaces, the horofunction boundary often matches or refines the visual, geodesic, or Satake compactifications, retaining group actions and equivariance properties (Arosio et al., 2020, Sato, 24 Mar 2025, Chu et al., 2022).
• Boundary dynamics: The fine structure of horofunction boundaries and their parts under group actions is closely related to the dynamics at infinity, random walks, and boundary extension of isometries (Lemmens, 2021, Lemmens et al., 2021).

In summary, horofunction compactification connects metric geometry, convexity, representation theory, and complex geometry, providing a powerful tool for the study of the large-scale and asymptotic properties of symmetric spaces, cones, and their generalizations (Chu et al., 2022, Lemmens et al., 2021, Lemmens, 2021, Walsh, 2014).