Horofunction Boundary Actions

• Horofunction boundary actions are induced by isometries and symmetries acting on normalized distance functions, encoding asymptotic strata, detour geometry, and Busemann versus non-Busemann behaviors.
• They utilize varied compactification techniques—including reduced boundaries, duality approaches, and vector-valued settings—to classify dynamics in both classical and non-metric scenarios.
• Applications extend from Hilbert and Teichmüller geometries to discrete groups, exposing rich boundary dynamics, rigidity in isometry actions, and deep interplays with convex-geometric duality.

Horofunction boundary actions are the actions induced on a horofunction compactification by isometries, group translations, or more general symmetries of the underlying metric or kernel. In its classical form, the compactification is built from normalized distance functions, but recent work also treats asymmetric metrics, vector-valued Cartan data, and even non-metric kernels. Across these settings, the boundary is not merely an abstract completion: it records asymptotic strata, Busemann versus non-Busemann behavior, detour geometry, dual-face data, and measure-theoretic cocycles that often control the dynamics of the original action (Walsh, 2014, Masai, 29 Dec 2025).

1. Foundational framework

For a possibly non-reversible metric space $(X,d)$ with basepoint $b$, the horofunction compactification is obtained by the embedding

$z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$

followed by closure in the topology of uniform convergence on bounded sets; the horofunction boundary is $X(\infty):=\overline{X}\setminus X$ (Walsh, 2014). A horofunction arising from an almost-geodesic is a Busemann point. On Busemann points one has the detour cost $H(\xi,\eta)$ and the symmetrized detour metric $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$; the finite-$\delta$ equivalence classes are the parts of the boundary, and these parts are preserved by isometries (Walsh, 2014).

The action mechanism is explicit. If $g:(X,d)\to (X',d')$ is an isometry, its extension to horofunctions is

$g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$

This gives a homeomorphic action on the compactification, preserves Busemann points and detour cost, and therefore preserves the boundary stratification by parts (Walsh, 2014). In finitely generated groups with word metric $d_S(g,h)=|g^{-1}h|_S$, the normalized embedding is

$b$0

and the induced left action is

$b$1

which descends to reduced quotients when such quotients are considered (Bodart et al., 2024).

Two reduction conventions coexist in the literature. One convention defines the reduced horoboundary by quotienting horofunctions by additive constants (Bodart et al., 2024). Another defines a reduced boundary by quotienting by bounded functions, via $b$2 (Bader et al., 2019). This distinction matters in nilpotent-group examples, where unreduced, constant-reduced, and bounded-reduced actions can behave differently.

The same action formula persists in higher-rank vector-valued settings. For a symmetric space $b$3, the vector-valued compactification uses

$b$4

for $b$5, projected to $b$6; the boundary action is

$b$7

and the associated vector-valued Busemann cocycle is $b$8 (Kim et al., 30 Mar 2026).

2. Hilbert geometry as a model case

In Hilbert geometry, the horofunction boundary admits an unusually explicit convex-geometric description and becomes a direct tool for isometry classification. For a bounded open convex set $b$9 in a finite-dimensional real vector space, the Hilbert distance is

$z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$0

where $z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$1 are collinear and ordered accordingly on the line through $z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$2. It splits as

$z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$3

with

$z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$4

(Walsh, 2014). This decomposition is the basic mechanism behind the boundary description.

The reverse-Funk boundary is elementary: $z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$5 extends continuously to $z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$6, and the reverse-Funk horofunctions are exactly

$z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$7

with convergence to $z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$8 equivalent to ordinary convergence to $z\mapsto \psi_z,\qquad \psi_z(x):=d(x,z)-d(b,z),$9. Every such horofunction is Busemann. Its parts are the relative interiors of proper extreme sets of $X(\infty):=\overline{X}\setminus X$0, and on each part the detour metric is the Hilbert metric of that extreme set (Walsh, 2014).

The Funk side is controlled by duality. Passing to the cone

$X(\infty):=\overline{X}\setminus X$1

one encodes the metric by the gauge $X(\infty):=\overline{X}\setminus X$2, with $X(\infty):=\overline{X}\setminus X$3. Funk horofunctions are then described through limiting slices of the dual cone $X(\infty):=\overline{X}\setminus X$4, and Funk Busemann points are parametrized by pairs $X(\infty):=\overline{X}\setminus X$5 with $X(\infty):=\overline{X}\setminus X$6 a proper extreme set of $X(\infty):=\overline{X}\setminus X$7 (Walsh, 2014). All Funk horofunctions are Busemann if and only if the set of extreme sets of the polar of $X(\infty):=\overline{X}\setminus X$8 is closed in the Painlevé–Kuratowski topology (Walsh, 2014).

The Hilbert boundary is assembled from these two asymmetrical pieces. A sequence converges to a Hilbert horofunction if and only if it converges in both the Funk and reverse-Funk geometries, and every Hilbert horofunction decomposes uniquely as

$X(\infty):=\overline{X}\setminus X$9

Moreover, $H(\xi,\eta)$0 is a Hilbert Busemann point if and only if $H(\xi,\eta)$1 is a Funk Busemann point. Compatibility is not automatic: if $H(\xi,\eta)$2 is a Funk Busemann point, then $H(\xi,\eta)$3 is a Hilbert Busemann point if and only if the extreme set $H(\xi,\eta)$4 of $H(\xi,\eta)$5 is contained in the exposed face of $H(\xi,\eta)$6 defined by $H(\xi,\eta)$7 (Walsh, 2014). This yields a simultaneous primal-dual stratification.

For Busemann points

$H(\xi,\eta)$8

the detour cost is additive: $H(\xi,\eta)$9 when $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$0 lie in the relative interior of the same extreme set $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$1 of $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$2 and $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$3, and $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$4 otherwise. Each part of the Hilbert Busemann boundary is therefore the direct product of two lower-dimensional Hilbert geometries with an $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$5-type metric (Walsh, 2014). Isometries must permute these parts and preserve their product structure.

This boundary description yields strong rigidity consequences. In polyhedral Hilbert geometries, one part with a trivial factor occurs for every vertex and every facet; analyzing how an isometry acts on these distinguished parts shows that any isometry either preserves the two families separately or interchanges them. If it preserves them, it is a collineation; if it interchanges them, this can happen only for simplices. Hence the only polyhedral Hilbert geometries with non-collineation isometries are simplices (Walsh, 2014). In the general case, extra isometries arise exactly from gauge-reversing maps on cones, which exist precisely for symmetric cones (Walsh, 2014).

3. Discrete groups, reduced boundaries, and rigidity versus richness

For finitely generated nilpotent groups, horofunction boundary actions display a sharp contrast between rigidity on visible asymptotic directions and abundance of non-Busemann phenomena. In discrete Heisenberg groups with the standard generators, every Busemann point is represented by a geodesic ray whose asymptotic direction is determined by a face of the associated $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$6-unit ball in the abelianization, and two Busemann points lie in the same orbit precisely when they have the same combinatorial asymptotic type. Consequently, the set of $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$7-orbits of Busemann points is finite, the set of Busemann points is countable, and the full horoboundary is nevertheless uncountable (Bodart et al., 2024). The uncountability is obtained through approximation by the ambient Heisenberg Lie group and its sub-Finsler asymptotic cone (Bodart et al., 2024).

The step-3 Cartan group behaves differently. It has continuum many Busemann points, produced by geodesic rays whose lower-central-series asymptotics retain a continuous parameter in the limit horofunction. As a byproduct, the action on the reduced horoboundary is nontrivial, disproving a conjecture of Bader–Finkelshtein, while the existence of continuum many Busemann points disproves a conjecture of Tointon–Yadin (Bodart et al., 2024). This establishes that higher-step nilpotent geometry can sustain richer reduced boundary dynamics than step-2 Heisenberg geometry.

A competing Heisenberg result uses the bounded-function reduction. For the discrete Heisenberg group $\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$8 with any finite symmetric generating set, the action on the corresponding reduced horoboundary is trivial (Bader et al., 2019). The mechanism is property EH: far from the identity, multiplying by a commutator changes word length by only a uniformly bounded amount. Since the Heisenberg commutator subgroup is central, bounded perturbations vanish in the reduced quotient, and every group element fixes every reduced boundary point (Bader et al., 2019). This does not contradict the Cartan example, because the Cartan paper uses the quotient by additive constants rather than bounded functions, and because the step-3 geometry supports genuinely new asymptotic parameters (Bodart et al., 2024, Bader et al., 2019).

The asymptotic-cone perspective clarifies this distinction. For polygonal sub-Finsler metrics on the real Heisenberg group, which arise as asymptotic cones of word metrics, the boundary action is

$\delta(\xi,\eta)=H(\xi,\eta)+H(\eta,\xi)$9

Smooth-direction horofunctions are fixed, finite orbits of Busemann points correspond exactly to edges of the defining polygon $\delta$0, and the reduced horoboundary is the orbit quotient, so the action there is trivial (Fisher et al., 2020). The same paper shows that the set of Busemann functions is homeomorphic to a circle, while the full horoboundary is larger; many horofunctions are not Busemann points (Fisher et al., 2020).

At the opposite dynamical extreme are Coxeter groups. For an irreducible non-spherical non-affine Coxeter group, the action on the horofunction boundary of the Cayley graph is minimal, topologically free, topologically amenable, and a strong boundary action (Ma et al., 7 Oct 2025). The proof uses contracting isometries with singleton attracting and repelling finite-difference classes, Myrberg points, and the identification of graph, combinatorial, Roller, and horofunction boundaries in the Coxeter setting (Ma et al., 7 Oct 2025). This places horofunction boundary actions squarely inside the boundary-dynamics toolkit of geometric group theory.

4. Enlarged boundaries and comparison with visual or ideal compactifications

A recurring theme is that horofunction compactifications are often strictly finer than more familiar boundaries. In infinite-dimensional real hyperbolic space $\delta$1, the horofunctions are precisely the functions

$\delta$2

for pairs $\delta$3 satisfying either $\delta$4 and $\delta$5, or $\delta$6. A horofunction is Busemann if and only if $\delta$7. Thus the Busemann points coincide with the classical ideal boundary, but the full horoboundary is strictly larger, and nets rather than sequences are required in general because $\delta$8 is not proper (Claassens, 2018). The additional boundary points arise from weak limits with escaped norm mass, a specifically infinite-dimensional phenomenon (Claassens, 2018).

In Teichmüller space with the Teichmüller metric, the horofunction compactification is finer than the visual compactification. For any proper, uniquely geodesic, straight space there is a continuous surjection

$\delta$9

sending a Busemann point $g:(X,d)\to (X',d')$0 to the ray $g:(X,d)\to (X',d')$1 (Azemar, 2021). Applied to Teichmüller space, this yields a path connected horoboundary and shows that Busemann points are not dense when $g:(X,d)\to (X',d')$2 for a closed surface of genus $g:(X,d)\to (X',d')$3 with $g:(X,d)\to (X',d')$4 marked points (Azemar, 2021). The same paper emphasizes that the mapping class group acts continuously on the Gardiner–Masur compactification, hence on the horofunction compactification, while a visual compactification based at a chosen basepoint generally does not support a continuous mapping class group action (Azemar, 2021).

For Teichmüller space of a surface with nonempty boundary equipped with the asymmetric arc metric,

$g:(X,d)\to (X',d')$5

the horofunction compactification is naturally homeomorphic to the Thurston compactification (Alessandrini et al., 2014). The boundary is therefore modeled by $g:(X,d)\to (X',d')$6, and boundary horofunctions are expressed explicitly through normalized length and intersection data,

$g:(X,d)\to (X',d')$7

This furnishes a concrete lamination model for horofunction boundary actions in the asymmetric Teichmüller setting (Alessandrini et al., 2014).

The comparison with Gromov compactifications can also be exact. For a proper geodesic Gromov hyperbolic space, the approaching geodesics property implies that the horofunction compactification is topologically equivalent to the Gromov compactification (Arosio et al., 2020). The weaker approaching geodesics property relative to a suitable family of rays is also sufficient (Arosio et al., 2020). In bounded strongly pseudoconvex domains with $g:(X,d)\to (X',d')$8 boundary, bounded convex domains with $g:(X,d)\to (X',d')$9, and bounded convex domains of finite D’Angelo type, the horofunction, Gromov, and Euclidean compactifications coincide (Arosio et al., 2020). In these cases, big and small horospheres in the sense of Abate coincide, and Julia’s lemma takes the horospherical form

$g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$0

for non-expanding self-maps (Arosio et al., 2020).

5. Duality, products, and higher-rank compactifications

In finite-dimensional normed spaces with polyhedral norm, the horofunction compactification can be converted into an explicit convex model. If $g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$1 is the unit ball and $g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$2 its dual unit ball, then the horofunction compactification is homeomorphic to $g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$3, and each boundary stratum indexed by a proper face $g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$4 consists of horofunctions

$g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$5

with $g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$6 (Ji et al., 2016). The relative interior of each face corresponds to one such family, so translations preserve face labels and act by shifting the transverse parameter $g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$7 (Ji et al., 2016). This makes the boundary action visible as an action on the face lattice of the dual polytope.

A broader duality theorem extends this picture to several Finsler-type settings. For product Kobayashi domains, Euclidean Jordan algebras with the spectral norm, and symmetric Hilbert geometries, the horofunction compactification is naturally homeomorphic to a closed dual unit ball, and each part of the boundary is mapped onto the relative interior of a boundary face (Lemmens et al., 2021). In these examples all horofunctions are Busemann points, so the detour partition of the boundary is entirely encoded by the face stratification of the dual ball (Lemmens et al., 2021). This suggests a general action principle: isometries should preserve not only the compactification but also the intrinsic partition into parts, hence the corresponding facial decomposition.

For products of negatively curved spaces, the maximum metric produces a boundary adapted to diagonal dynamics. If $g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$8 is a proper $g(\xi)(x'):=\xi(g^{-1}(x'))-\xi(g^{-1}(b')).$9 space, then the horofunction boundary of $d_S(g,h)=|g^{-1}h|_S$0 splits into a singular part and a regular part; the regular part is parametrized by

$d_S(g,h)=|g^{-1}h|_S$1

through horofunctions of the form

$d_S(g,h)=|g^{-1}h|_S$2

(García et al., 2018). For an infinite quasi-convex subgroup $d_S(g,h)=|g^{-1}h|_S$3, there exists a maximal open subset $d_S(g,h)=|g^{-1}h|_S$4 of the horofunction boundary such that the diagonal action on

$d_S(g,h)=|g^{-1}h|_S$5

is properly discontinuous and cocompact (García et al., 2018). The construction uses an equivariant extension of nearest-point projection to the diagonal and identifies the regular off-diagonal boundary with the space of parameterized geodesics in $d_S(g,h)=|g^{-1}h|_S$6 (García et al., 2018).

In higher-rank symmetric spaces, the compactification can be made vector-valued. For $d_S(g,h)=|g^{-1}h|_S$7, $d_S(g,h)=|g^{-1}h|_S$8, and $d_S(g,h)=|g^{-1}h|_S$9, one considers

$b$00

projects to $b$01, and forms the compactification $b$02 (Kim et al., 30 Mar 2026). The action is

$b$03

and the boundary cocycle is

$b$04

The partial flag manifold $b$05 embeds in $b$06, but $b$07 is larger and is analytically more convenient for Patterson’s construction (Kim et al., 30 Mar 2026). The paper defines shadows $b$08, proves a shadow lemma, and shows that for strongly irreducible $b$09-transverse groups satisfying the divergence hypothesis there is a unique $b$10-Patterson–Sullivan measure on $b$11, the action is ergodic, and the measure is fully supported on the conical limit set (Kim et al., 30 Mar 2026).

6. Beyond metric compactifications

Recent work has generalized horofunction theory from metrics to arbitrary kernels $b$12. The associated distance on $b$13 is

$b$14

and the generalized horofunction attached to $b$15 is

$b$16

If $b$17 separates points of $b$18 and $b$19 and the symmetrized distances are finite, then

$b$20

is continuous and injective, producing a horo-compactification of $b$21 (Masai, 29 Dec 2025). If a group acts diagonally invariantly on $b$22 and $b$23, the induced boundary action is

$b$24

and it extends continuously to the compactifications (Masai, 29 Dec 2025). Under north-south dynamics, boundary fixed points encode generalized translation lengths, exactly as in metric horofunction theory (Masai, 29 Dec 2025).

The Teichmüller application of this framework is structurally significant. For

$b$25

one obtains two asymmetric distances. One of them recovers the Teichmüller metric, while the other,

$b$26

is a horofunction counterpart to Teichmüller distance (Masai, 29 Dec 2025). The same paper identifies the classical horofunction boundary of Teichmüller distance with the Gardiner–Masur compactification and identifies the compactification given by

$b$27

with the Thurston compactification (Masai, 29 Dec 2025). This shows that boundary actions can be constructed from the same asymmetric kernel in more than one way.

A max-plus analogue appears in the directed landscape. There the Martin boundary coincides with the horofunction boundary, horofunctions are exactly eternal solutions with a spatial growth rate, and the minimal Martin boundary is given by the Busemann functions (Rassoul-Agha et al., 20 May 2026). Every eternal solution is a max-plus convex combination of countably many Busemann functions, while every horofunction admits a representation using at most two Busemann functions with a common growth rate (Rassoul-Agha et al., 20 May 2026). Because not all horofunctions are Busemann functions, the Martin boundary is strictly larger than its minimal part (Rassoul-Agha et al., 20 May 2026). A plausible implication is that horofunction boundary actions extend naturally beyond metric compactifications to semigroup and max-plus harmonic settings, provided one has an analogue of a Busemann cocycle and a harmonic fixed-point equation.