This is the accepted version of the journal article: Porti, Joan; García Galvez, María Teresa. «Actions on products of CAT(-1)- spaces». Geometriae Dedicata, In press 2023. 26 pàg. 2023. This version is available at https://ddd.uab.cat/record/275306 under the terms of the license ACTIONS ON PRODUCTS OF CAT(−1) SPACES TERESA GARCÍA AND JOAN PORTI Abstract. We show that for X a proper CAT(−1) space there is a maximal open subset of the horofunction compactification of X × X, with respect to the maximum metric, that compactifies the diagonal action of an infinite quasi-convex group of the isometries of X. We also consider the product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper CAT(−1) spaces. 1. Introduction The action of a discrete group of isometries Γ on the ideal boundary of a proper CAT(−1) space X has a dynamical decomposition ∂∞ X = ΩΓ ⊔ ΛΓ , where ΛΓ is the limit set and ΩΓ is the domain of discontinuity [6]. In addition, if Γ is quasi-convex, then the action on X ∪ ΩΓ is also properly discontinuous and cocompact, so ΩΓ compactifies the action of Γ on X [18]. This dynamical decomposition of the visual boundary may not hold for CAT(0) spaces: there may be no uniqueness of maximal discontinuity do- mains, or there may be no discontinuity domain at all, even if the limit set is proper. We mention the work of Papasoglu and Swenson [17], or Kapovich, Leeb, and Porti [12], in the context of symmetric spaces. In this paper we consider the case of the product of two proper CAT(−1) spaces, which is a CAT(0) space, see the work on products by Geninska [10] and by Link [14]. As a motivating example, consider a cocompact fuchsian group Γ < Isom(H2 ) acting diagonally on H2 × H2 . The ideal boundary of H2 is ∂∞ H2 ∼ = S 1 , so the visual boundary of the product is the spherical join of two circles, ∂∞ (H2 × H2 ) ∼ = S 1 × S 1 × [0, π/2]/ ∼, where ∼ is the relation that collapses each subset {∗}×S 1 ×{0} or S 1 ×{∗}×{π/2} to a point. The diagonal action on the ideal boundary preserves the sets S 1 × S 1 × {θ} for each θ ∈ [0, π/2], so finding a domain of discontinuity amounts to find a do- main of discontinuity for the diagonal action on S 1 ×S 1 . This is not possible because the action on S 1 × S 1 has a dense orbit [15, Thm. 3.6.1], hence it has empty domain of discontinuity (even if the limit set is S 1 × S 1 × {π/4}). Notice that the visual compactification of H2 × H2 is the horofunction com- pactification with respect to the product metric, or ℓ2 metric. Instead, here we work with the ℓ∞ or maximum metric, which happens to be better suited for those compactifications. Date: May 12, 2023. First author supported by grant BES-2013-065701 and second author by Marı́a de Maeztu Program for Centers and Units of Excellence (CEX2020-001084-M). Both authors partially supported by grant FEDER-Meic MTM2015–66165–P. 1 2 TERESA GARCÍA AND JOAN PORTI For X1 and X2 two proper CAT(−1) spaces, we denote the ideal or Gro- mov boundary of X1 ×X2 with respect to the max metric by ∂∞ max (X ×X ). 1 2 max In Proposition 3.13 we show that ∂∞ (X1 × X2 ) is homeomorphic to the join of the boundaries of each factor. In particular, the ideal boundaries for both metrics ℓ2 and ℓ∞ are homeomorphic, but their compactifications are not equivalent, since the identity does not extend continuously to the com- pactifications. The max compactification is adapted to diagonal actions, as it allows to find an ideal subset where the diagonal action is properly dis- continuous and which compactifies the action. The main theorem of this paper is: Theorem 1.1. Let X be a proper CAT(−1) space and Γ an infinite quasi- convex group of isometries of X. There exists an open set Ωmax Γ ⊂ ∂∞max (X × X) such that: (a) The diagonal action of Γ on X × X ∪ ΩmaxΓ is properly discontinuous and cocompact. (b) Ωmax Γ max (X × X) where the diagonal is the largest open subset of ∂∞ action is properly discontinuous. When Γ acts cocompactly on X, the theorem has been proved in [9]. To prove Theorem 1.1 we show that the nearest point projection from X × X to the diagonal extends continuously to a map on ∂∞ max (X × X) with image in the visual compactification of the diagonal. The ideal boundary ∂∞ max (X × X ) decomposes in two parts defined in 1 2 Section 3, regular and singular. The regular part ∂∞ max (X ×X ) 1 2 reg consist of points that correspond to the maximum of two Busemann functions, one on each factor, and it is homeomorphic to ∂∞ X1 ×∂∞ X2 ×R (Proposition 3.10). The singular part ∂∞ max (X × X ) 1 2 sing consists of points that are Busemann functions in one of the factors and it is homeomorphic to the disjoint union ∂∞ X1 ⊔ ∂∞ X2 (Proposition 3.9). In a CAT(0) space the limit set ΛΓ is the set of accumulation points in the ideal boundary of an orbit and it is independent of the choice of the orbit. In our setting, since the max metric is not CAT(0), the set of accumulation points of an orbit depends on the orbit, so we consider the large limit set, consisting of accumulation points of any orbit. For a diagonal action it turns out that the large limit set is contained in the regular part of the boundary and that Ωmax Γ is the complement of the closure of the large limit set. In the particular case in which Γ is a cocompact group, the set Ωmax Γ is naturally homeomorphic to the set of parameterized geodesics in one factor, as shown in [9]. This max metric is a Finsler metric. Bordifications through Finsler met- rics of symmetric spaces have been used by Kapovich and Leeb [11] to ob- tain a characterization of Anosov representations. In a product of CAT(−1) spaces, this corresponds to the ℓ1 metric. The max compactification is very convenient for diagonal actions, but it would be interesting to see in what other situations it is useful. For Γ an infinite hyperbolic group, we consider ρ1 and ρ2 two quasi-convex representations in the respective group of isometries of CAT(−1) spaces X1 and X2 , and their product action ρ1 × ρ2 on X1 × X2 : an element γ in Γ ACTIONS ON PRODUCTS OF CAT(−1) SPACES 3 maps (x1 , x2 ) ∈ X1 × X2 to (ρ1 (γ)x1 , ρ2 (γ)x2 ). Since the max metric on X1 × X2 is not CAT(0), the accumulation set of an orbit may depend on the orbit. The union of all possible accumulation sets is called the large limit set and it is denoted by Λρ1 ×ρ2 ⊂ ∂∞ max (X × X ). In analogy to the diagonal 1 2 case, it is reasonable to ask under what conditions the large limit set Λρ1 ×ρ2 of the product action also remains inside the regular part of the boundary. Proposition 1.2. Let X1 and X2 be proper CAT(−1) spaces, Γ an infinite hyperbolic group, and ρ1 : Γ → Isom(X1 ), ρ2 : Γ → Isom(X2 ) two quasi- convex representations. The following are equivalent: max (X × X ) . (a) Λρ1 ×ρ2 ⊂ ∂∞ 1 2 reg (b) There exists C > 0 depending on o ∈ X1 and o′ ∈ X2 such that |d1 (ρ1 (γ)o, ρ1 (γ ′ )o) − d2 (ρ2 (γ)o′ , ρ2 (γ ′ )o′ )| < C, for all γ ∈ Γ. (c) The length spectrun is the same: τ (ρ1 (γ)) = τ (ρ2 (γ)) for all γ ∈ Γ, where τ (ρi (γ)) denotes the translation length of ρi (γ). When item (b) holds we say that ρ1 and ρ2 are coarsely equivalent. If both representations ρ1 and ρ2 are coarsely equivalent and cocompact, then the spaces X1 and X2 are almost-isometric. This means that there exists an almost-isometry between the spaces, which is a quasi-isometry with mul- tiplicative constant one. This almost-isometry allows to construct a coarse equivariant map between the regular parts of the ideal boundaries of X1 ×X1 max (X ×X ) of Theorem 1.1 is mapped and X1 ×X2 , so that the open set in ∂∞ 1 1 to an open set Ωmax Γ ⊂ ∂∞ max (X × X ) with good properties: 1 2 Theorem 1.3. Let X1 and X2 be proper CAT(−1) spaces, Γ a hyperbolic group, and ρ1 : Γ → Isom(X1 ), ρ2 : Γ → Isom(X2 ) two cocompact discrete representations. If ρ1 and ρ2 are coarsely equivalent, then there exists an open subset Ωmax Γ ⊂ ∂∞max (X × X ) such that the product action of Γ on 1 2 X1 × X2 ∪ Ωmax Γ is properly discontinuous and cocompact. 2. Preliminaries A metric space is said to be proper if all its closed balls are compact, and geodesic if any two points can be joined by a geodesic segment. A CAT(−1) space X is a geodesic metric space where triangles are thinner than comparison triangles in the hyperbolic plane. Similarly, a CAT(0) space satisfies the same condition placing the comparison triangles in the Euclidean plane. In particular, CAT(−1) spaces are also CAT(0) spaces. A reference for these spaces is for instace [4]. Two rays c(t) and c′ (t) in a metric space are said to be asymptotic if there exists C < ∞ such that d(c(t), c′ (t)) ≤ C for any t ≥ 0. The visual boundary ∂∞ X of a metric space X is the set of equivalent classes of asymptotic rays. In a proper CAT(0) space X = X ∪ ∂∞ X can be given a topology (the cone topology, see [4]) such that both X and ∂∞ X are compact. The space X is denoted the visual compactification. A discrete group action on a topological space X is properly discontinu- ous if every compact subset intersects finitely many of its translates. For isometric actions on proper metric spaces, this is equivalent to the fact that every point has an open neighborhood which intersects only finitely many 4 TERESA GARCÍA AND JOAN PORTI of its translates. The action is cocompact if there exists a compact subset K ⊂ X whose translates cover X. For Γ a discrete group of isometries of a proper CAT(0) space, the limit set ΛΓ is defined as the set of accumulation points of an orbit in ∂∞ X and it is independent of the orbit. For a CAT(−1) space X, the complement of ΛΓ in ∂∞ X is the domain of discontinuity ΩΓ and Γ acts properly discontinuously on ΩΓ [6]. A subset S ⊂ X is quasi-convex if an ε-neighborhood of S ∩ X contains its quasi-convex hull (the union of segments between points in S), for some ε > 0. A group Γ of the isometries of a CAT(−1) space X is quasi-convex if it acts properly discontinuously on X and any orbit is quasi-convex. A map between metric spaces f : X → Y is a quasi-isometric embedding if there are constants A ≥ 1 and C ≥ 0 satisfying that for all x1 , x2 ∈ X: 1 dX (x1 , x2 ) − C ≤ dY (f (x1 ), f (x2 )) ≤ A dX (x1 , x2 ) + C. A The map f : X → Y is a quasi-isometry if it is a quasi-isometric embedding that is coarsely onto, namely if for each y ∈ Y there exists x ∈ X with: dY (f (x), y) ≤ C, for some C. An almost-isometry is a quasi-isometry with multiplicative constant A = 1. A quasi-convex group of isometries of a proper CAT(−1) space X is hy- perbolic and finitely generated. Moreover, the orbit map: Γ→X γ 7→ γo is a quasi-isometric embedding for any o ∈ X and it extends to an equivariant homeomorphism (which is also Lipschitz and quasi-conformal) from ∂∞ Γ to its limit set ΛΓ [3]. The action of a quasi-convex group on X ∪ΩΓ is properly discontinuous [6] and cocompact [18]. 3. The max compactification Let (X1 , d1 ) and (X2 , d2 ) be two proper CAT(−1) spaces, consider the product space X1 × X2 equipped with the max metric dmax , or ℓ∞ metric: dmax ((x, y), (x′ , y ′ )) = max{d1 (x, x′ ), d2 (y, y ′ )} for any (x, y), (x′ , y ′ ) ∈ X1 × X2 . The metrics dmax and the product metric dℓ2 ((x, y), (x′ , y ′ )) = d1 (x, x′ )2 + d2 (y, y ′ )2 p are comparable so they induce the same topology in X1 ×X2 . For X1 and X2 proper geodesic spaces, (X1 × X2 , dmax ) is also a proper geodesic space [16, Prop.2.6.6]. In this section we compute its horofunction compactification. For a proper metric space X, let C∗ (X) denote space of continuous func- tions on X up to additive constants, equipped with the topology of uniform convergence on compact subsets. The Gromov or horofunction compactifi- cation of X is the closure in C∗ (X) of the image of the map ι : X → C∗ (X) x 7→ [d(x, ·)], ACTIONS ON PRODUCTS OF CAT(−1) SPACES 5 see [2]. We denote the horofunction compactification of X by X. The ideal boundary is the set X \ ι(X) and it is denoted by ∂∞ X. For a proper metric space both X and ∂∞ X are compact and metrizable spaces. Remark 3.1. Fixing a base point o ∈ X, the sequence [d(xn , ·)] converges to a class of functions [f ] ∈ C∗ (X) if, and only if, the sequence of correspond- ing normalized distance functions d(xn , ·) − d(xn , o) converges to f − f (o) uniformly on all balls B(o, r). In addition, C∗ (X) is homeomorphic to the subspace of continuous functions on X satisfying f (o) = 0. Definition 3.2. A horofunction h is a continuous function on X such that its class [h] belongs to ∂∞ X. Remark 3.3. A class of horofunctions is called an ideal point and it is denoted it by ξ. The horofunction h in the class ξ satisfying h(o) = 0 is denoted by hoξ . Notation 3.4. When we say that a sequence (xn )n converges to an ideal point ξ in the horofunction compactification, xn → ξ, we mean that for a base point o ∈ X the corresponding sequence of normalized distance functions converges uniformly on compact subsets to the horofunction hoξ . The level sets of a horofunction are called horospheres and the sublevel sets, horoballs. Notice that two horofunctions in the same equivalence class differ by a constant and share the same set of horospheres and horoballs. The horofunctions of a proper CAT(0) space are Busemann functions: Definition 3.5 (See [1] or [2]). A Busemann function in a metric space (X, d) is a function defined as: z 7→ lim d(c(t), z) − t t→+∞ for some geodesic ray c(t) in X. In a proper CAT(0) space X, given a point o ∈ X and an ideal point ξ ∈ ∂∞ X there is a unique ray c(t) such that c(0) = o and its associated Busemann function is in the class ξ. This Busemann function is denoted by βξo (z) = lim d(c(t), z) − t, t→+∞ The horofunction compactification and the visual compactification of a proper CAT(0) space are equivalent [4, Cor. 8.20]. Lemma 3.6 (cf. [4]). For a CAT(0) space X: (i) If σ : [0, +∞) → X is a ray in the class ξ ∈ ∂∞ X, then βηo (σ(s)) converges to +∞ if η ̸= ξ and to −∞ if η = ξ. (ii) For any p, q ∈ X, βηp − βηq is a constant function. The Gromov or horofunction compactification of (X1 × X2 , dmax ) is de- max max (X × X ). We choose noted by X1 × X2 and its ideal boundary by ∂∞ 1 2 ′ ′ a base point O = (o, o ) with o ∈ X1 and o ∈ X2 . As a representative of a class of normalized distance functions, we have the function: ′ dO max ((x, y), ·) = dmax ((x, y), ·) − dmax ((x, y), (o, o )) = max{d1 (x, ·), d2 (y, ·)} − max{d1 (x, o), d2 (y, o′ )}. 6 TERESA GARCÍA AND JOAN PORTI Then, by Remark 3.1, [dmax ((xn , yn ), ·)] → ξ ∈ ∂∞ max (X × X ) if and only 1 2 if dO O O max ((xn , yn ), ·) → hξ , where hξ is the horofunction in the class ξ that satisfies hOξ (O) = 0. Given a diverging sequence (xn , yn ) ⊂ (X1 × X2 , dmax ), we distinguish two cases, up to a subsequence: (I) either |d1 (xn , o) − d2 (yn , o′ )| → ∞, (II) or |d1 (xn , o) − d2 (yn , o′ )| remains bounded. Notice that if one of d1 (xn , o) or d2 (yn , o′ ) is bounded, then we are in the first case, as we assume that (xn , yn ) diverges. Proposition 3.7. Let (X1 , d1 ) and (X2 , d2 ) be proper CAT(−1) metric spaces. Let (xn , yn ) be a diverging sequence in (X1 × X2 , dmax ). (I) If |d1 (xn , o) − d2 (yn , o′ )| → ∞, then, up to subsequence and up to permuting X1 and X2 , there exists ξ ∈ ∂∞ X1 such that lim dO ((xn , yn ), (z, z ′ )) = βξo (z). n→∞ max (II) If |d1 (xn , o) − d2 (yn , o′ )| remains bounded, then, up to a subsequence, there exist ξ ∈ ∂∞ X1 and ξ ′ ∈ ∂∞ X2 such that ′ ′′ lim dO o o max ((xn , yn ), (z, z )) = max{βξ (z), βξ ′ (z ) − C}, n→∞ for some constant C ∈ R. Proof. We prove case (II), the proof for case (I) being similar. For each n, denote Cn = d1 (xn , o) − d2 (yn , o′ ) and assume that Cn ≥ 0. Then dmax ((xn , yn ), (o, o′ )) = d1 (xn , o) = d2 (yn , o′ ) + Cn and ′ dO max ((xn , yn ), (z, z )) = dmax ((xn , yn ), (x, y)) − dmax ((xn , yn ), (o, o′ )) = max{d1 (xn , x) − d1 (xn , o), d2 (yn , y) − d2 (yn , o′ ) − Cn }. Both sequences xn and yn subconverge to an ideal point, and since Cn is bounded we are done. □ In the remaining of the section, (X1 , d1 ) and (X2 , d2 ) denote proper CAT(−1) metric spaces, as in Proposition 3.7. Definition 3.8. We define the singular part of the ideal boundary as the ′ subset of ideal points with a representative of the form βξo (z) or βξo′ (z ′ ), i.e. case (I) in Proposition 3.7. We denote it by ∂∞ max (X × X ) 1 2 sing . The regular part of the ideal boundary is its complement, namely the ′ subset of ideal points with a representative of the form max{βξo (z), βξo′ (z ′ ) + C} with C ∈ R, i.e. case (II). We denote it by ∂∞ max (X × X ) . 1 2 reg Using that the set of Busemann functions in one factor is naturally iden- tified to the boundary of this factor, we have: Proposition 3.9. There is a natural homeomorphism max φsing : ∂∞ (X1 × X2 )sing −→ ∂∞ X1 ⊔ ∂∞ X2 ACTIONS ON PRODUCTS OF CAT(−1) SPACES 7 that consists in associating to a Busemann function that takes values only in the first (second) factor of X1 × X2 the same Busemann function viewed as a point of the first (second) summand in ∂∞ X1 ⊔ ∂∞ X2 . For the regular part, notice that we can get rid of the additive constant in Proposition 3.7 by changing the base point. Thus regular points are the ′ classes modulo constant of the functions max{βξp (z), βξp′ (z ′ )} for all p ∈ X1 , p′ ∈ X2 , ξ ∈ ∂∞ X1 and ξ ′ ∈ ∂∞ X2 . Proposition 3.10. For each choice of base point (o, o′ ) ∈ X1 × X2 there is a natural homeomorphism max h : ∂∞ (X1 × X2 )regi −→ ∂∞ X1 × ∂∞ X2 × R φreg (1) ′ ′ max{βξp (z), βξp′ (z ′ )} 7→ (ξ, ξ ′ , βξp (o) − βξp′ (o′ )). Remark 3.11. If we fix p = o and p′ = o′ , then homeomorphism (1) can be written as: max (X × X ) φreg : ∂∞ 1 2 reg −→ ∂∞ X1 × ∂∞ X2 × R (2) ′ max{βξ (z), βξo′ (z ′ ) − C} 7→ (ξ, ξ ′ , C) o where C ∈ R. ′ Proof of Prop. 3.10. Notice that limt→+∞ max{βξp (c(t)), βξp′ (c′ (t))} = −∞ if and only if c(+∞) = ξ and c′ (+∞) = ξ ′ ; otherwise this limit is +∞, by Lemma 3.6. Thus ξ and ξ ′ in the construction of φreg are uniquely determined, and it follows easily that φreg is well defined and injective. In addition, surjectivity of φreg and continuity of φ−1 reg follow easily from construction and the properties of Busemann functions (Lemma 3.6). To prove continuity of φreg , as ideal boundaries are metrizable, we use ′ sequences. Let max{βξon (z), βξon′ (z ′ ) − Cn } be a sequence that converges to ′ max{βηo (z), βηo′ (z ′ )−C}. The third coordinate of φreg in (1) is clearly contin- uous, hence Cn → C. By compactness of ∂∞ Xi , up to subsequence ξn → ξ∞ and ξn′ → ξ∞′ . By injectivity of φ , ξ reg ′ ′ ∞ = η and ξ∞ = η and we get conti- nuity. □ Remark 3.12. Observe that a sequence (xn , yn ) converges to (ξ, ξ ′ , C) if, and only if, xn → ξ, yn → ξ ′ , and d1 (xn , o) − d2 (yn , o′ ) → C. Let Join(∂∞ X1 , ∂∞ X2 ) denote the topological join of ∂∞ X1 and ∂∞ X2 . Propositions 3.9 and 3.10 can be improved: Proposition 3.13. There is a natural homeomorphism ∂ max (X1 × X2 ) ∼ ∞ = Join(∂∞ X1 , ∂∞ X2 ). Proof. In view of Propositions 3.9 and 3.10 and Remark 3.11, we have to prove the following claim: for sequences (ξn ) in ∂∞ X1 , (ξn′ ) in ∂∞ X2 , and (Cn ) in R we have ξn → ξ, ξn′ → ξ ′ , and Cn → +∞ as n → ∞ if and only if the function ′ (z, z ′ ) 7→ max{βξon (z), βξon′ (z ′ ) − Cn } converges to (z, z ′ ) 7→ βξo (z) uniformly on compact subsets of X1 ×X2 . Notice that we do no require convergence on (ξn′ ). We also need the symmetric ′ claim when Cn → −∞, and after replacing max{βξon (z), βξon′ (z ′ ) − Cn } by 8 TERESA GARCÍA AND JOAN PORTI ′ max{βξon (z)+Cn , βξon′ (z ′ )} (a different function in the same equivalence class) but the proof is symmetric. To prove the claim assume first that Cn → +∞. Take as compact set the ball centered at (o, o′ ): B(o, R) × B(o′ , R). Since Busemann functions ′ are 1-Lipschitz and as we chose normalizations so that βξon (o) = βξon′ (o′ ) = ′ 0, for Cn ≥ 2R we have max{βξon (z), βξon′ (z ′ ) − Cn } = βξon (z) for (z, z ′ ) ∈ B(o, R) × B(o′ , R). Here uniform convergence of βξon on compact subsets follows from the horosphere compactification of X1 . Next assume Cn ∈ ′ [−R, R]. Here max{βξon (z), βξon′ (z ′ ) − Cn } has a converging subsequence to ′ max{βξo (z), βξo′ (z ′ ) − C}, uniformly on compact subsets. Using that the Busemann functions have slope -1 in rays pointing to the ideal point, we ′ see that the limit max{βξo (z), βξo′ (z ′ ) − C} cannot be expressed a Busemann ′ function in a single factor, βξo or βξo′ . □ 4. Diagonal actions Let Γ be an infinite quasi-convex group of isometries of a proper CAT(−1) space X. In this section we consider the diagonal action of Γ on X × X: Γ×X ×X →X ×X (γ, x, y) 7→ (γx, γy). The diagonal action extends continuously to the ideal boundary of the max compactification. The following is straightforward: max (X × X) is given Lemma 4.1. The diagonal action on the points of ∂∞ by: o γ[βξo ] = [βγξ ] −1 γ[max{βξo , βξo′ − C}] = [max{βγξ o o , βγξ o ′ − C + βξ ′ (γ o) − βξo (γ −1 o)}]. Remark 4.2. Under the identification in Remark 3.11 the diagonal action maps a singular point ξ to γξ, and a regular point (ξ, ξ ′ , C), to (γξ, γξ ′ , C + βξo (γ −1 o) − βξo′ (γ −1 o)). In this section we prove that there is an open subset Ωmax Γ ⊂ ∂∞max (X ×X) where the diagonal action of Γ is properly discontinuous and cocompact. In Subsection 4.1 we prove that the nearest point projection of X × X to the max diagonal ∆ ⊂ X × X extends continuously to X × X . In Subsection 4.2 we use this projection to show that there exist a proper domain of discon- tinuity Ωmax Γ ⊂ ∂∞max (X × X). Furthermore we see that the action on the whole X × X ∪ Ωmax Γ is properly discontinuous and cocompact, and that Ωmax Γ is the largest open set of the boundary that satisfies these conditions. 4.1. Extending the projection to the diagonal. The nearest point pro- jection from X × X to the diagonal for the max distance is given by the midpoint: π: X × X → ∆ (3) (x, y) 7→ (m, m), ACTIONS ON PRODUCTS OF CAT(−1) SPACES 9 where ∆ is the diagonal in X × X: ∆ = {(x, x) | x ∈ X}, and m is the midpoint of the geodesic segment joining x and y. By con- struction, π is continuous and equivariant. In this section we extend it continuously to a map max max π̃ : X × X →∆ = ∆ ∪ ∆∞ , max max where ∆ is the closure of ∆ in X × X , and ∆∞ = {(ξ, ξ) | ξ ∈ ∂∞ X}, denotes the diagonal in ∂∞ X × ∂∞ X. For this purpose, we consider the decomposition max ∂∞ max (X × X) = ∂∞ (X × X)sing ⊔ φ−1 reg (∆∞ × R) ⊔ Ω max where Ωmax = φ−1 max reg ((∂∞ X × ∂∞ X \ ∆∞ ) × R) ⊂ ∂∞ (X × X)reg ., and φreg is the homeomorphism in Proposition 3.10. In [9] the projection is extended continuously to a map Ωmax → ∆. Following [9], the extension uses that Ωmax is naturally homeomorphic to the set G of parameterized geodesics in X (with the topology of uniform convergence on compact sets) through the map: φ : G −→ Ωmax g 7→ lim (g(n), g(−n)) n→∞ Via this identification, by [9] the projection extends continuously to G → ∆ (4) g 7→ (g(0), g(0)) max (X × X) Thus it remains to extend it to φreg (∆∞ × R) and to ∂∞ sing . Definition 4.3. The extended projection max max ∼ π̃ : X × X →∆ =X is defined by (3) and (4) on X × X ∪ Ωmax . max (X × X) On φreg (∆∞ × R) it is the projection to ∆∞ , and on ∂∞ ∼ sing = ∼ ∂∞ X ⊔ ∂∞ X it is the identification ∂∞ X = ∆∞ . Remark 4.4. We have an equivariant homeomorphism φ′ = φreg ◦ φ, given by: φ′ : G → ((∂∞ X × ∂∞ X) \ ∆∞ ) × R g 7→ (g(+∞), g(−∞), Cg ), where: o o Cg = lim d(g(n), o) − d(g(−n), o) = βg(−∞) (g(0)) − βg(+∞) (g(0)). n→∞ Therefore, a geodesic g corresponds to a point (g(+∞), g(−∞), Cg ) ∈ ((∂∞ X × ∂∞ X) \ ∆∞ ) × R 10 TERESA GARCÍA AND JOAN PORTI which in its turn corresponds to the regular point: o o [max{βg(+∞) , βg(−∞) − Cg }] ∈ Ωmax . To prove the continuity of π̃, we use of that the Gromov product ex- tends continuously to the boundary of a proper CAT(−1) space [5, Propo- sition 3.4.2]. The Gromov product of two points x, y ∈ X with respect to a base point o ∈ X is defined as: 1 (x|y)o = [d(x, o) + d(y, o) − d(x, y)] . 2 Given ξ and ξ ′ two points in the visual boundary of a proper CAT(−1), the Gromov product is defined as: lim(xi |yj )o = (ξ|ξ ′ )o , i,j for any sequences xi → ξ, yj → ξ ′ . Let g be a geodesic in X, the Gromov product of the ideal points g(+∞) and g(−∞) with respect to a base point o, can be written in terms of Busemann functions as: 1 h g(0) g(0) i (g(+∞)|g(−∞))o = βg(+∞) (o) + βg(−∞) (o) . 2 The Gromov product for ideal points satisfies: (ξ|ξ ′ )o = +∞ if and only if ξ = ξ ′ , see [5]. Similarly two sequences xi , yj have the same limit iff: (xi |yj )o = +∞. max Theorem 4.5. The map π̃ : X × X → X is continuous and equivariant. Proof. The equivariance follows from naturality. To prove the continuity, we have also shown in [9] that π̃ restricted to X × X ∪ Ωmax is continuous, but it remains to be proved in ∂∞ max (X × X) \ Ωmax . We have to check two cases. (I) Firstly, we shall see that the image of a sequence of points (xn , yn ) in X × X that converges to an ideal point, either in the singular part or in the diagonal of the regular part of the boundary, converges to the image of this ideal point. (II) Secondly, we shall check that the image of a sequence of ideal points that converges to an ideal point either in the diagonal of the regular part or in the singular part of the boundary, converges to the image of the ideal point. Along this proof, mn denotes the midpoint of the segment joining xn and yn . Case (I). Consider a sequence (xn , yn ) in X × X converging to an ideal point. We distinguish two subcases: either (a) the limit of the sequence is a singular point, or (b) the limit is a point in the diagonal of the regular part. Subcase (a). Suppose, up to permuting factors, that the sequence con- verges to a singular point in the boundary of the first factor: (xn , yn ) → [βξo ]. Therefore, xn → ξ and d(xn , o) − d(yn , o) → +∞. By the triangle inequal- ity: d(mn , o) ≥ d(mn , yn ) − d(yn , o), and using the definition of the Gromov product, we have: 1 (xn |mn )o ≥ [d(xn , o) − d(yn , o)] . 2 ACTIONS ON PRODUCTS OF CAT(−1) SPACES 11 Henceforth (xn |mn )o → +∞, and by the properties of the Gromov product, xn and mn have the same limit. Subcase (b). Now suppose that the sequence converges to a diagonal point in the regular part of the boundary: (xn , yn ) → [max{βξo , βξo + C}]. In this case xn → ξ, yn → ξ and d(xn , o) − d(yn , o) → −C. Using the definition of the Gromov product again and reorganizing terms, we have: 1 (5) 2(xn |mn )o = (xn |yn )o + d(mn , o) + [d(xn , o) − d(yn , o)] . 2 Notice that 12 [d(xn , o) − d(yn , o)] is uniformly bounded and that (xn |yn )o → +∞, since both xn and yn converge to the same point. From (5) we deduce that (xn |mn )o → +∞, which implies that mn → ξ. Case (II). Next we deal with a sequence of regular ideal points of the form [max{βξon , βξon′ + Cn }] in ∂∞ max (X × X) with limit either a regular point in the diagonal, subcase (a), or a singular point, subcase (b). From now on, gn denotes the geodesic corresponding to a point [max{βξon , βξon′ +Cn }] under the identification Ωmax ∼ = G. Subcase (a). Suppose that the sequence converges to a regular diagonal point: [max{βξon , βξon′ + Cn }] → [max{βξo , βξo + C}]. In this case ξn → ξ, ξn′ → ξ and Cn → C. For each n, we consider a sequence of points xk in X such that xk → ξn . Using the fact that the Gromov product extends continuously to the boundary of a CAT(−1) space, and the definition of Busemann function, we write: 1 (gn (0)|ξn )o = lim (gn (0)|xk )o = lim [d(gn (0), o) + d(xk , o) k→∞ k→∞ 2 1h g (0) i −d(gn (0), xk )] = d(gn (0), o) + βξnn (o) . 2 Similarly, taking a sequence yk in X for each n, with yk → ξn′ : 1h g (0) i (gn (0)|ξn′ )o = d(gn (0), o) + βξ′n (o) . 2 n Adding both equalities above we obtain: 1 g (0) 1 g (0) (6) (gn (0)|ξn )o + (gn (0)|ξn′ )o = d(gn (0), o) + βξnn (o) + βξ′n (o) 2 2 n = d(gn (0), o) + (ξn |ξn′ )o . By compactness of X, after passing to a subsequence we may assume that gn (0) → η ∈ X. Then, since (ξn |ξn′ )o → (ξ|ξ)o = +∞, by (6) we have that (η|ξ)o = +∞. So ξ = η and gn (0) → ξ. Subcase (b). Next we suppose that the sequence converges to a singular point, a Busemann function in the second factor (up to permutation of factors): [max{βξon , βξon′ + Cn }] → [βξo′ ] with ξn → ξ, ξn′ → ξ ′ and since βξo′ is a Busemann function in the second factor, Cn → +∞. Similarly to the preceding case, for each n: 1h i (gn (0)|ξn′ )o = d(gn (0), o) − βξon′ (gn (0)) , 2 12 TERESA GARCÍA AND JOAN PORTI 1 d(gn (0), o) − βξon (gn (0)) ,  (gn (0)|ξn )o = 2 and we can combine both equalities to get: 1 1 (gn (0)|ξn′ )o = d(gn (0), o) − βξon′ (gn (0)) 2 2 1 o 1 = (gn (0)|ξn )o + βξn (gn (0)) − βξon′ (gn (0)) 2 2 1 = (gn (0)|ξn )o + Cn . 2 Here we have used that Cn = βξn (gn (0)) − βξon′ (gn (0)) by Remark 4.4. Now, o since (gn (0)|ξn )o ≥ 0 and Cn → +∞ we have that (gn (0)|ξn′ )o → +∞ and gn (0) → ξ ′ . □ 4.2. The ideal domain Ωmax Γ . Let Γ be an infinite quasi-convex group of isometries of X. We denote by ΛΓ its limit set, which is the set of accumulation points of any orbit in ∂∞ X, and by ΩΓ = ∂∞ X \ΛΓ its domain of discontinuity. The action of Γ on X ∪ ΩΓ is properly discontinuous and cocompact [6,18]. Next we show that the diagonal action of Γ on the inverse image under the projection π̃ of X ∪ ΩΓ is also properly discontinuous and cocompact: Theorem 4.6. Let X be a proper CAT(−1) space and let Γ ⊂ Isom(X) be an infinite quasi-convex group. The diagonal action of Γ on π̃ −1 (X ∪ ΩΓ ) is properly discontinuous and cocompact. max Proof. Besides being continuous and equivariant, π̃ : X × X → X is proper, since it is a continuous map from a compact to a Hausdorff space. In [18] it is shown that for a Dirichlet domain D ⊂ X its closure in D ⊂ X is a compact set that satisfies: S ⊂ X ∪ ΩΓ , (i) D (ii) γ∈Γ γD = X ∪ ΩΓ , and (iii) for every compact K ⊂ X ∪ ΩΓ , |{γ ∈ Γ | γK ∩ K ̸= ∅}| < ∞. Therefore π̃ −1 (D) satisfies the corresponding properties for the action on π̃ −1 (X ∪ ΩΓ ). This proves the theorem. □ Now, let Ωmax Γ be the intersection of π̃ −1 (X ∪ΩΓ ) with the ideal boundary max of X × X : Ωmax Γ = π̃ −1 (X ∪ ΩΓ ) ∩ ∂∞max (X × X), and let ∆Γ be the subset of the diagonal in ∂∞ X × ∂∞ X that corresponds to the limit points of the action of Γ on X: ∆Γ = {(ξ, ξ) ∈ ∂∞ X × ∂∞ X with ξ ∈ ΛΓ }. Remark 4.7. Via the homeomorphism in Proposition 3.13 that identifies max (X × X) with Join(∂ X, ∂ X): ∂∞ ∞ ∞ max ∼ Ω Γ = ((∂∞ X × ∂∞ X) \ ∆Γ ) × R ∪ Ω1 ∪ Ω2 , Γ Γ where ΩiΓ denotes ΩΓ viewed in the factor ∂∞ Xi , for each i = 1, 2. Observe that when Γ is a cocompact group Ωmax Γ is just the set Ωmax of the previous subsection. ACTIONS ON PRODUCTS OF CAT(−1) SPACES 13 In Proposition 4.10 we shall show that Ωmax Γ is the largest open set of the boundary where the diagonal action is properly discontinuous. But first, let us study the limit set of this action on ∂∞max (X × X). Since (X × X, d max ) is not CAT(0), the accumulation set of each orbit depends on the orbit. We define the large limit set of the diagonal action as the union of all the accumulation sets of orbits on ∂∞max (X × X): [ max Λ= Γ(x, y) ∩ ∂∞ (X × X) (x,y)∈X×X Lemma 4.8. For Λ and ∆Γ as above, φreg (Λ) = ∆Γ × R. Proof. First, observe that the limit of any sequence (γn x, γn y) that converges to the ideal boundary is contained in ∆Γ × R. Indeed, by the triangle inequality, |d(γn x, o) − d(γn y, o)| ≤ d(x, y), so the limit is a regular point. Furthermore, if γn x → ξ then γn y → ξ since d(γn x, γn y) = d(x, y), i.e. the sequences γn x and γn y stay within a bounded distance. Therefore the limit point lies in ∆Γ × R. Finally we show that any point (ξ, ξ, C) for ξ ∈ ΛΓ belongs to the limit set. Take any sequence γn such that γn o → ξ. Let ξ ′ be an accumulation point for γn−1 o and x, y ∈ X satisfying βξo′ (y) − βξo′ (x) = C. Then it follows easily that (ξ, ξ, C) is the limit of the sequence (γn x, γn y). □ Remark 4.9. The large limit set Λ is not closed but notice that the com- plement of Ωmax Γ is its closure Λ. From this remark we easily deduce: Proposition 4.10. The set ΩmaxΓ max (X × X) is the largest open subset of ∂∞ max such that the action of Γ on X × X ∪ ΩΓ is properly discontinuous. 5. Examples In this section we consider some examples for the max compactification of diagonal actions. The first one is the diagonal action of a cocompact group of isometries of a riemannian manifold. The second one is the action of convex cocompact Kleinian groups on Hn × Hn . Finally we describe an example of the max compactification of a diagonal action on the product of two trees. This is also an example where the nearest point projection to the diagonal is not a fibration. 5.1. Compact riemannian manifolds. As in the previous section, let π : X × X → ∆ denote the nearest point projection to the diagonal for the max distance. By (3), the fibre π −1 (a, a) is the set of pairs (x, y) ∈ X × X such that a is the midpoint of the segment joining x and y. If X = H2 , then the fibre Fa = π −1 (a, a) is: Fa = {(x, sa x) | x ∈ H2 } where sa x is the symmetric point of x with respect to a. Then Fa ∼= H2 for any a. The boundary at infinity of the fibre Fa is the set of parameterized geodesics with g(0) = a, so: ∂ max Fa = {g | g(0) = a} ∼ ∞ = (Ta H2 )1 14 TERESA GARCÍA AND JOAN PORTI For S = H2 /Γ a compact hyperbolic surface, the max compactification of (H2 × H2 )/Γ is the fibration by closed disks of S = H2 /Γ, so: max (H2 × H2 )/Γ ∼ = U S, where U S = {(x, v) ∈ T S | |v| ≤ 1}. The same is true for a Cartan-Hadamard manifold X of dimension n and sectional curvature ≤ −1: ∂∞ Fx is identified with the unit tangent sphere at x, (Tx X)1 ∼ = S n−1 , and the fibre over each point of the diagonal is a closed disk. If M = X/Γ is a compact manifold, then the compactification of (X × X)/Γ with respect to the max metric is homeomorphic to the fibration by closed disks of M = X/Γ: max (X × X)/Γ ∼ = U M, where U M = {(x, v) ∈ T M | |v| ≤ 1}. 5.2. Convex cocompact Kleinian groups. Let Γ < Isom+ (Hn ) be a discrete torsion free subgroup, that is convex cocompact. Assume that M = Hn /Γ is not compact, then it has finitely many ends and its compactification consists in adding a compact conformal (n − 1) manifold Nin−1 = Ωi /Γi , where Ωi is a connected component of the discontinuity domain Ω = ∂∞ Hn \ ΛΓ . max The compactification (Hn × Hn )/Γ is the union of the fibration by compact balls on M and a finite collection of products of the conformal ideal manifolds with intervals, Nin−1 × R, where R = R ∪ {−∞, +∞} ∼ = [0, 1]. n−1 To understand how these products Ni × [0, 1] are attached, we consider a diverging geodesic ray in r : [0, +∞) → M , corresponding or a point in ∂∞ M . For each t ∈ [0, +∞), the fibre π −1 (r(t)) is a compactified hyperbolic space Hn , we aim to understand how they fit with a compactified R when t → ∞. Assume that r is a ray in Hn . Points in π −1 (r(t)) are of the form (expr(t) (v), expr(t) (−v)), for some v ∈ Tr(t) Hn . To compare different fibers, let V be a parallel vector field along r that is unitary. Let θ ∈ [0, 2π) be the angle between r′ (t) and V (t), which is constant by parallelism. Then every point of π −1 (r(t)) is written as (expr(t) (s V (t)), expr(t) (−s V (t))) for some V as above and s ∈ R≥0 . Proposition 5.1. Given V as above and s ∈ R≥0 , r(0) r(0) lim (expr(t) (s V (t)), expr(t) (−s V (t))) = max{βr(+∞) , βr(+∞) − 2d} t→+∞ where d ∈ R is defined by the relation (7) tanh d = cos θ tanh s and θ is the (constant) angle between r′ and V . The relation (7) means that d is the signed distance from r(t) to the orthogonal projection of expr(t) (s V (t)) to the ray r, see Figure 1. This proposition explains how the Hn in the fibre are attached to a segment in the limit: the whole Hn is projected orthogonally to the geodesic containing r, and, by continuous extension, Hn is projected to R = R ∪ {−∞, +∞}. ACTIONS ON PRODUCTS OF CAT(−1) SPACES 15 expr(t) (s V (t)) s r θ d r(t) Figure 1. The triangle for Equality (7) in Proposition 5.1 Proof. As the distance from expr(t) (±s V (t)) to r(t) is s, we know that the limit is a maximum of Busemann functions centered at r(+∞). Therefore we only have to compute the limit lim d(expr(t) (s V (t)), r(0)) − d(expr(t) (−s V (t)), r(0)). t→+∞ Set d± (t) = d(expr(t) (±s V (t)), r(0)). By the hyperbolic cosine formula: cosh d+ (t) = cosh s cosh t − cos θ sinh s sinh t, cosh d− (t) = cosh s cosh t + cos θ sinh s sinh t. Hence cosh d+ (t) cosh s − cos θ sinh s lim ed+ (t)−d− (t) = lim = t→+∞ t→+∞ cosh d− (t) cosh s + cos θ sinh s By taking logarithms: 1 − cos θ tanh s 1 − tanh d lim d+ (t) − d− (t) = log = log = −2d. □ t→+∞ 1 + cos θ tanh s 1 + tanh d 5.3. Constant valence trees. Let T be the tree of valence 4 and edges of unit length, it is the Cayley graph of F2 , the free group on two generators. Let ∆ ⊂ T × T denote the diagonal. If we consider the nearest point projection to the diagonal, then the fibres over different points of ∆ might be different. In fact, given (a, a) ∈ ∆ there are three possible fibres, which are topologically not equivalent, depending on whether a ∈ T is a vertex, a midpoint of an edge or a generic point in an edge. Theorem 5.2. For T the tree of valence four and edges of unit length, the fibres over ∆ are of one of the following mutually exclusive types: (1) Generic fibres. For a ∈ T a generic point in an edge, the fibre over (a, a) is a tree of valence 4. The metric in the fibre depends on the distance from a to its nearest vertex in T , L with 0 < L < 1/2. Along any path through (a, a), the length of consecutive edges alter- nate between 2L and L′ = 1 − 2L. The point (a, a) is the midpoint of an edge of length 2L. (2) Midpoint fibres. For a ∈ T a midpoint of an edge, the fibre over (a, a) is a tree of constant valence 10. All edges have length 1 and the point (a, a) is the midpoint of an edge. This is the limit of the previous case when L → 1/2. 16 TERESA GARCÍA AND JOAN PORTI Figure 2. Case 1 Figure 3. Case 2 (3) Vertex fibres. For a ∈ T a vertex, the fibre over (a, a) is a tree of constant valence 10, except in the point (a, a), which is a valence 12 vertex. All edges have length 1. This is the limit of the generic case when L → 0, taking into account that there are four fibres approaching to the base point, one for each edge in T incident to a. Figure 4. Case 3 Proof. The fibre of (a, a) is the set of pairs (x, y) ∈ T ×T so that the midpoint of the segment xy is a. To reach all such a pairs, we start from a and consider pairs of paths obtained by moving at speed one along T and pointing away from a. The first requirement is that we start in different directions, ie for ACTIONS ON PRODUCTS OF CAT(−1) SPACES 17 small times the points (x, y) of the pair are already different. When one of the points reach a vertex, we consider all possible continuations along different edges. This construction provides a graph structure on the fibre: when x and y move along an edge this yields an edge of the fibre, when at least one of them reaches a vertex of T , this yields a vertex of the fibre. This fibre is in fact a tree, because we can orient each edge of the initial tree T so that it points in the direction opposite to a (when a lies in the interior of an edge, we split this edge along a). This yields an orientation of the edges of the fibre, so that each edge points away from (a, a). In addition, at every vertex only edge points to this vertex, the other edges point away, hence it is a tree. We describe the tree for the fibre of a vertex in T , case 3, the other two cases are a follow from similar arguments. Let a be a vertex and denote by (x, y) a point in the fibre over (a, a). For each x in a edge incident to a, there are three possibilities for y, such that a is the midpoint of x and y, one from each of the three remaining edges. In total there are 4 edges incident to a, so there are 4·3 = 12 edges incident to (a, a). Next we follow x and y along two edges, always satisfying d(x, a) = d(y, a), until both x and y are two vertices v and v ′ . Then, for x there are three new possibilities, one for each new edge incident to v, and if we follow the path along one of the edges there are three possibilities for y, such that d(x, a) = d(y, a), one for each new edge incident to the vertex v ′ . So there are 3 · 3 + 1 = 10 edges incident to (v, v ′ ). This pattern repeats, given rise to a tree with valence 10 in all vertices except the base point. The distance between two consecutive vertices, for instance (a, a) and (v, v ′ ), is dmax ((a, a), (v, v ′ )) = max{d(a, v), d(a, v ′ )} = d(a, v) = 1, so the edges have length 1. □ Remark 5.3. Let T be again the tree of valence 4, also the Cayley graph of F2 . The projection T ×T → ∆ ∼ = T induces a map from (T ×T )/F2 to T /F2 , which is a wedge of two circles. The fibres are trees, as in Theorem 5.2, and the tree depends on the point on the wedge T /F2 : Case 3 for the vertex of the wedge, Case 2 for the midpoints of the edges, and Case 1 for the remaining (generic) points. The ideal boundary of each fibre in the max compactification is its boundary at infinity as a tree, which is a Cantor set in any case. The previous considerations of course apply to free groups of rank n and their Cayley graphs. 6. Product actions Let (X1 , d1 ) and (X2 , d2 ) be two proper CAT(−1) spaces and let Γ be an infinite hyperbolic group with ρ1 : Γ → Isom(X1 ), ρ2 : Γ → Isom(X2 ) two quasi-convex representations (by ρi quasi-convex we mean that it has finite kernel and that ρi (Γ) is a discrete quasiconvex group). In this section we study under what conditions the large limit set Λρ1 ×ρ2 of the product action Γ × X1 × X2 → X1 × X2 (γ, x, y) 7→ (ρ1 (γ)x, ρ2 (γ)y) 18 TERESA GARCÍA AND JOAN PORTI lies inside the regular part of the boundary. We shall see that asking the large limit set to lie in the regular part of the boundary is in fact a very restrictive condition, which is related to the marked length spectrum con- jecture. Indeed, in Subsection 6.1 we prove the following proposition: max Proposition 6.1. Let X1 , X2 be proper CAT(−1) spaces and X1 × X2 the horofunction compactification with respect to dmax . Let Γ be an infinite hyperbolic group and ρ1 : Γ → Isom(X1 ), ρ2 : Γ → Isom(X2 ) two quasi- convex representations. The following are equivalent: (a) Λρ1 ×ρ2 ⊂ ∂∞max (X × X ) 1 2 reg . (b) ρ1 ≃C.E. ρ2 . (c) τ (ρ1 (γ)) = τ (ρ2 (γ)) for all γ ∈ Γ. Here τ denotes the translation length of an isometry, see Definition 6.6 below, so condition (c) means that both representations have the same trans- lation lengths. Condition (b) requires the following definition: Definition 6.2. The representations ρ1 , ρ2 are said to be coarsely equiva- lent if there exists C > 0 such that: |d1 (ρ1 (γ)o, ρ1 (γ ′ )o) − d2 (ρ2 (γ)o′ , ρ2 (γ ′ )o′ )| ≤ C for some o ∈ X1 and o′ ∈ X2 , and for all γ, γ ′ ∈ Γ. When ρ1 and ρ1 are coarsely equivalent, we write written ρ1 ≃C.E. ρ2 If the representations are coarsely equivalent for some base-points o ∈ X1 and o′ ∈ X2 then they are coarsely equivalent for any choice of base-points in X1 and X2 . We see later in Lemma 6.12 how the bound depends on the choice of base-points. Definition 6.3. A K-almost-isometry between two metric spaces is a map f : X1 → X2 such that |d(x, y) − d(f (x), f (y))| ≤ K, ∀x, y ∈ K, and X2 lies in the K-neighborhood of f (X1 ). If ρ1 and ρ2 are coarsely equivalent and cocompact, then the spaces X1 and X2 are equivariantly almost-isometric. In this case it is possible to find a subset Ωmax Γ max (X × X ) where the product action is properly dis- of ∂∞ 1 2 continuous and cocompact. In Subsection 6.2 we use the existence of this almost-isometry between X1 and X2 and its extension to the ideal bound- aries of the spaces to prove the following theorem: max Theorem 6.4. Let X1 , X2 be proper CAT(−1) spaces and X1 × X2 the horofunction compactification with respect to dmax . Let Γ be a hyperbolic group and ρ1 : Γ → Isom(X1 ), ρ2 : Γ → Isom(X2 ) two cocompact discrete representations with finite kernel. If ρ1 and ρ2 are coarsely equivalent, then there exists a subset Ωmax Γ ⊂ ∂∞max (X × X ) where the product action of Γ 1 2 is properly discontinuous and cocompact. 6.1. Regular limit sets and coarsely equivalent representations. Let Γ be a group acting on a space X with two metrics d1 and d2 that are Γ- invariant. ACTIONS ON PRODUCTS OF CAT(−1) SPACES 19 Definition 6.5. Two metrics d1 and d2 on a space X are coarsely equivalent if there exists a constant C ≥ 0 such that for all x, y ∈ X, |d1 (x, y) − d2 (x, y)| ≤ C. Definition 6.6. The metrics d1 , d2 have the same marked length spectrum with respect to the action of Γ if τ1 (γ) = τ2 (γ) for all γ ∈ Γ, where τi (γ) is the translation length of γ for di defined by di (x, γ n (x)) τi (γ) = lim n→∞ n for any x ∈ X. The equivalence of both definitions for hyperbolic groups follows from results of Furman [8] and Krat [13]. Theorem 6.7. (Furman, Krat) Let Γ be a hyperbolic group acting on itself with two left invariant metrics d1 , d2 which are quasi-isometric to a word metric by the identity map. Then d1 and d2 are coarsely equivalent if and only if d1 and d2 have the same marked length spectrum. We are interested in a hyperbolic group acting on two proper CAT(-1) spaces (X1 , d1 ), (X2 , d2 ) via quasi-convex representations ρ1 , ρ2 into their respective groups of isometries. A hyperbolic group Γ acts on itself by left translations. Moreover, fixing oi ∈ Xi , each representation ρi induces an orbit map from Γ to the target space Xi for i = 1, 2: Oi : Γ → Xi γ 7→ ρi (γ)oi These orbit maps induce left invariant metrics dΓi in Γ by: dΓi (γ, γ ′ ) = di (ρi (γ)oi , ρi (γ ′ )oi ) so that for i = 1, 2 (Γ, dΓi ) are Γ invariant metric spaces. Moreover, since (Xi , di ) are proper CAT(−1) spaces and the representations are quasi-convex, these metrics are quasi-isometric to a word metric by the identity map, see [3]. Remark 6.8. The metrics dΓ1 and dΓ2 are coarsely equivalent if and only if ρ1 ≃C.E. ρ2 . Indeed, in both cases the condition to be satisfied is that there exists a constant C such that |d1 (ρ1 (γ)o1 , ρ1 (γ ′ )o1 ) − d2 (ρ2 (γ)o2 , ρ2 (γ ′ )o2 )| ≤ C for some o1 ∈ X1 , o2 ∈ X2 and for all γ, γ ′ ∈ Γ. Using Remark 6.8, Theorem 6.7 yields: Proposition 6.9. Let X1 , X2 be proper CAT(−1) spaces. Let Γ be an infinite hyperbolic group and ρ1 : Γ → Isom(X1 ), ρ2 : Γ → Isom(X2 ) two quasi-convex representations. Then: ρ1 ≃C.E. ρ2 ⇔ τ (ρ1 (γ)) = τ (ρ2 (γ)) for all γ ∈ Γ. 20 TERESA GARCÍA AND JOAN PORTI Next we prove the equivalence (a) ⇔ (b) in Proposition 6.1. Recall that the large limit set of the product action ρ1 × ρ2 is the union of accumulation sets of all orbits on ∂∞max (X × X ): 1 2 [ max Λρ1 ×ρ2 = {(ρ1 (γ)x, ρ2 (γ)y) | γ ∈ Γ} ∩ ∂∞ (X1 × X2 ). (x,y)∈X1 ×X2 Recall also that the regular part of the ideal boundary can be identified with the product of the ideal boundaries of each factor and R: ∂∞ (X1 × X2 )reg ∼= ∂X1 × ∂X2 × R. Fixing a base point (o, o′ ) ∈ X1 × X2 , a sequence (xn , yn ) ⊂ X1 × X2 converges to a point (ξ, ξ ′ , C) in the regular part if: xn → ξ ∈ ∂X1 , yn → ξ ′ ∈ ∂X2 , and d1 (xn , o) − d2 (yn , o′ ) → C ∈ R. max (X × X ) Proposition 6.10. If Λρ1 ×ρ2 ⊂ ∂∞ 1 2 reg then ρ1 ≃C.E. ρ2 . Proof. Suppose that ρ1 and ρ2 are not coarsely equivalent. This means that there is a sequence γn in Γ such that |d1 (o, ρ1 (γn )o) − d2 (o′ , ρ2 (γn )o′ )| is un- bounded. By definition of singular point, this means that (ρ1 (γn )o, ρ2 (γn )o′ ) accumulates in the singular part, which is a contradiction. □ For the implication (b) ⇒ (a) we need a couple of lemmas. The first one is a direct consequence of the triangle inequality: Lemma 6.11. Let x, y, z and t be four points in a metric space (X, d). Then: |d(x, y) − d(z, t)| ≤ d(x, z) + d(y, t). Lemma 6.12. If ρ1 ≃C.E. ρ2 then for any x ∈ X1 , y ∈ X2 : |d1 (ρ1 (γ)x, ρ1 (γ ′ )x) − d2 (ρ2 (γ)y, ρ2 (γ ′ )y)| ≤ C + 2(d1 (x, o) + d2 (y, o′ )) for all γ, γ ′ in Γ and for a C depending only on o and o′ . Proof. We add and subtract d1 (ρ1 (γ)o, ρ1 (γ ′ )o) and d2 (ρ2 (γ)o′ , ρ2 (γ ′ )o′ ) and apply the triangle inequality: |d1 (ρ1 (γ)x, ρ1 (γ ′ )x) − d2 (ρ2 (γ)y, ρ2 (γ ′ )y)| ≤ |d1 (ρ1 (γ)x, ρ1 (γ ′ )x) − d1 (ρ1 (γ)o, ρ1 (γ ′ )o)| + |d1 (ρ1 (γ)o, ρ1 (γ ′ )o) − d2 (ρ2 (γ)o′ , ρ2 (γ ′ )o′ )| + |d2 (ρ2 (γ)o′ , ρ2 (γ ′ )o′ ) − d2 (ρ2 (γ)y, ρ2 (γ ′ )y)| Next we find a bound for each summand of the right-hand side. By Lemma 6.11: |d1 (ρ1 (γ)x, ρ1 (γ ′ )x) − d1 (ρ1 (γ)o, ρ1 (γ ′ )o)| ≤ d1 (ρ1 (γ)x, ρ1 (γ)o) + d1 (ρ1 (γ ′ )x, ρ1 (γ ′ )o) = 2d1 (x, o) and |d2 (ρ2 (γ)o′ , ρ2 (γ ′ )o′ ) − d2 (ρ2 (γ)y, ρ2 (γ ′ )y)| ≤ d2 (ρ2 (γ)o′ , ρ2 (γ)y)) + d2 (ρ2 (γ ′ )o′ , ρ2 (γ ′ )y) = 2d2 (y, o′ ). ACTIONS ON PRODUCTS OF CAT(−1) SPACES 21 In addition, by assumption we have: |d1 (ρ1 (γ)o, ρ1 (γ ′ )o) − d2 (ρ2 (γ)o′ , ρ2 (γ ′ )o′ )| ≤ C, so the result follows. □ Remark 6.13. Observe that Lemma 6.12 implies that the definition of coarse equivalence does not depend on the orbit. max (X × X ) . Proposition 6.14. If ρ1 ≃C.E. ρ2 then Λρ1 ×ρ2 ⊂ ∂∞ 1 2 reg Proof. We want to show that sequences of the form (ρ1 (γn )x, ρ2 (γn )y) ac- cumulate in the regular part; equivalently |d1 (ρ1 (γn )x, o) − d2 (ρ2 (γn )y, o′ )| is bounded, so every accumulation point of the sequence is in the regular part. Applying Lemma 6.12 with x = o, y = o′ , γ = γn and γ ′ = Id we get that |d1 (ρ1 (γn )o, o)−d2 (ρ2 (γn )o′ , o′ )| ≤ C. Then, it follows from the triangle inequality that |d1 (ρ1 (γn )x, o)−d2 (ρ2 (γn )y, o′ )| ≤ d1 (x, o)+d2 (y, o′ )+C. □ 6.2. Compactification of product actions. In this section we consider ρ1 : Γ → Isom(X1 ), ρ2 : Γ → Isom(X2 ) two discrete cocompact coarsely equivalent representations. We shall show that, as in the diagonal case, there exists an open subset Ω ⊂ ∂∞ max (X × X ) of the ideal boundary such that 1 2 the product action on X1 × X2 ∪ Ω is properly discontinuous and cocompact. Lemma 6.15. If ρ1 : Γ → Isom(X1 ), ρ2 : Γ → Isom(X2 ) are coarsely equiv- alent cocompact representations, then there exists an equivariant almost- isometry f : X1 → X2 . Proof. Since the action is cocompact on both spaces X1 and X2 , each of these spaces is equivariantly almost-isometric to any orbit of Γ. The condition of coarse equivalence implies that the orbits of Γ in X1 are equivariantly almost-isometric to the orbits of Γ in X2 . □ Remark 6.16. The almost-isometry f is not unique. max (X × X ) such that Γ acts properly discontinuously and To find Ω ⊂ ∂∞ 1 2 cocompactly on X1 × X2 ∪ Ω, consider f : X1 → X2 the almost-isometry of Lemma 6.15 and use the map Id × f : X1 × X1 → X1 × X2 to translate the properties of the diagonal action ρ1 × ρ1 on X1 × X1 to the product action ρ1 × ρ2 on X1 × X2 . The almost-isometry f of Lemma 6.15 has an almost-inverse f −1 : X2 → X1 such that: d1 (f −1 (f (x1 )), x1 ) ≤ K and d2 (f (f −1 (x2 )), x2 ) ≤ K, for all x1 ∈ X1 and x2 ∈ X2 . Since quasi-isometries between CAT(−1) spaces extend to homeomorphisms of the boundaries, f extends to an equi- variant homeomorphism: f∞ : ∂∞ X1 → ∂∞ X2 , whose inverse is the extension of the almost-isometry f −1 . Remark 6.17. All the choices of almost-isometries f : X1 → X2 extend to the same map f∞ : ∂∞ X1 → ∂∞ X2 . 22 TERESA GARCÍA AND JOAN PORTI For i = 1, 2, let max (X × X )) φi : (∂∞ 1 i reg → ∂∞ X1 × ∂∞ Xi × R z 7→ (ξi (z), ηi (z), hi (z)) be the homeomorphism of Proposition 3.10. Choose o ∈ X1 and f (o) ∈ X2 as base points to compute h1 and h2 as in Proposition 3.10: h1 ([(x, x′ ) 7→ max{β(x), β ′ (x′ )}] = β(o) − β ′ (o), h2 ([(x, x′ ) 7→ max{β(x), β ′′ (x′′ )}] = β(o) − β ′′ (f (o))), where β and β ′ are Busemann functions on X1 and β ′′ on X2 . The domain of discontinuity of the diagonal action is Ω1 = φ−1 1 ((∂∞ X1 × ∂∞ X1 \ ∆∞ ) × R), where ∆∞ denotes the diagonal of ∂∞ X1 . For the action on X1 × X2 define Ω2 = φ−1 2 ((∂∞ X1 × ∂∞ X2 \ ∆f∞ ) × R), where ∆f∞ is graph of f∞ : ∆f∞ = {(ξ, η) ∈ ∂∞ X1 × ∂∞ X2 | η = f∞ (ξ)}. By Remark 4.9, Ω1 = ∂∞ max (X × X ) \ Λ , where Λ denotes the large 1 1 1 1 limit set of the diagonal action. For Ω2 we also have: Remark 6.18. Let Λ2 denote the large limit set of the (ρ1 × ρ2 )-action. max (X × X ) \ Λ . Then Ω2 = ∂∞ 1 2 2 As Remark 4.9, this remark follows from the fact that Λ2 ∼ = ∆f∞ × R (the proof of this equality is similar to Lemma 4.8). We next prove that Ω2 is the set Ωmax Γ in the statement of Theorem 6.4. For this purpose we consider a map F : Ω1 → Ω2 defined as follows. Every z ∈ Ω1 can be written as z = lim (g(−n), g(n)), n→+∞ for a unique geodesic g in X1 . This construction yields a homeomorphism between the set of bi-infinite geodesics in X1 and Ω1 . Next, if φ1 (z) = (ξ(z), η(z), h1 (z)), then define F (z) by  φ2 (F (z)) = ξ(z), f∞ (η(z)), h2 (F (z)) , where  h2 (F (z)) = lim sup d1 (g(n), o) − d2 (f (g(−n)), f (o)) . n→+∞ Thus, for any bi-infinite geodesic g in X1 ,  F lim (g(−n), g(n)) = lim (g(−nk ), f (g(nk ))) n→+∞ k→+∞ for some diverging subsequence (nk )k . Notice that the map F : Ω1 → Ω2 may be non continuous and depends on the choice of f . Lemma 6.19. Let K be the constant of almost isometry of f . Then: (i) |h2 (F (z)) − h1 (z)| ≤ K, ∀z ∈ Ω1 . (ii) For i = 1, 2, if z, z ′ ∈ Ωi satisfy ξi (z) = ξi (z ′ ) and ηi (z) = ηi (z ′ ), then, ∀γ ∈ Γ, hi (γz) − hi (γz ′ ) = hi (z) − hi (z ′ ). ACTIONS ON PRODUCTS OF CAT(−1) SPACES 23 (iii) |h2 (F (γz)) − h2 (γF (z))| ≤ 4K, ∀z ∈ Ω1 , ∀γ ∈ Γ. Proof. (i) Write z ∈ Ω1 as the limit z = limn→+∞ (g(−n), g(n)) for a (unique) geodesic g in Ω1 . Then h1 (z) = lim d(g(−n), o) − d(g(n), o), n→+∞  h2 (F (z)) = lim sup d g(−n), o) − d(f (g(n)), f (o) . n→+∞ From these expressions we get |h2 (F (z)) − h1 (z)| ≤ lim sup |d(f (g(n)), f (o)) − d(g(n), o)| ≤ K. n→+∞ (ii) We prove it for i = 2, as the proof for i = 1 is analogous. By Lemma 4.1: f (o) h2 (γz) − h2 (z) = βξo2 (z) (γ −1 o) − βη2 (z) (γ −1 f (o)). As we assume ξ2 (z) = ξ2 (z ′ ) and η2 (z) = η2 (z ′ ), assertion (ii) is proved. (iii) We write:   h2 (F (γz)) − h2 (γF (z)) = h2 (F (γz)) − h1 (γz) + h1 (γz) − h1 (z)   + h1 (z) − h2 (F (z)) + h2 (F (z)) − h2 (γF (z)) = (I) + (II) + (III) + (IV ). The terms (I) and (III) are bounded in absolute value by K by item (i). By Lemma 4.1: (II) = h1 (γz) − h1 (z) = βξo+ (γ −1 o) − βξo− (γ −1 o), f (o) (IV ) = h2 (F (z)) − h2 (γF (z)) = −βξo+ (γ −1 o) + βf∞ (ξ− ) (γ −1 f (o)). Hence f (o) (8) (II) + (IV ) = βf∞ (ξ− ) (f (γ −1 o)) − βξo− (γ −1 o). For r : [0, +∞) → X1 the geodesic ray with r(0) = o that converges to ξ− : (9) βξo− (γ −1 o) = lim d1 (r(t), γ −1 o) − d1 (r(t), o). t→+∞ On the other hand, f ◦r : [0, +∞) → X2 is a quasi-geodesic that converges to f∞ (ξ− ). Since the visual compactification and the compactification by ho- rofunctions are the same for a CAT(−1)-space, there is a diverging sequence (tk ) → +∞ such that f (o) (10) βf∞ (ξ− ) (f (γ −1 o)) = lim d2 (f (r(tk )), f (γ −1 o)) − d1 (f (r(tk )), f (o)). k→+∞ Since f is a K-almost isometry, it follows from (8), (9) and (10) that |(II) + (IV )| ≤ 2K. □ Lemma 6.20. Let z ∈ Ω1 and y ∈ Ω2 be such that ξ1 (z) = ξ2 (y), f∞ (η1 (z)) = η2 (y), and h1 (z) = h2 (y). Then |h1 (γz) − h2 (γy)| ≤ 6K, ∀γ ∈ Γ. 24 TERESA GARCÍA AND JOAN PORTI Proof. It is a consequence of the following three inequalities: |h1 (γz) − h2 (F (γz))| ≤ K, |h2 (F (γz)) − h2 (γF (z))| ≤ 4K, |h2 (γF (z)) − h2 (γy)| = |h2 (F (z)) − h2 (y)| = |h2 (F (z)) − h1 (z)| ≤ K. Here we have used Lemma 6.19, item (i) for the first line, item (iii) for the second, and items (ii) and (i) for the last one. □ Proposition 6.21. The action of Γ on Ω2 is properly discontinuous and cocompact. Proof. We prove proper discontinuity by showing that no two points in Ω2 are dynamically related. Recall that two points x, y in a metric space Z are dynamically related by Γ if there exist a sequences (zn )n in Z and (γn )n in Γ such that zn → x, γn → ∞, and γn zn → y, see [7]. Proper discontinuity is equivalent to the property that any two points (possibly equal) are not dynamically related. By contradiction, we assume y∞ and y∞ ′ in Ω are dynamically related, 2 and we shall show that two points in Ω1 are dynamically related. Namely, assume that there exists a sequence (yn )n in Ω2 and a diverging sequence (γn )n in Γ such that yn → y∞ ∈ Ω2 and γn yn → y∞ ′ ∈ Ω . For each n ∈ 2 N let zn ∈ Ω1 be such that ξ1 (zn ) = ξ2 (yn ), f∞ (η1 (zn )) = η2 (yn ), and h1 (zn ) = h2 (yn ) (we have defined φi = (ξi , ηi , hi )). Since ϕ1 and ϕ2 are homeomorphisms, zn → z∞ ∈ Ω1 . On the other hand, the coordinates ξ1 (γn zn ) and η1 (γn zn ) also converge and it remains to bound |h1 (γn zn )|: by Lemma 6.20 |h1 (γn zn ) − h2 (γn yn )| ≤ 6K and h2 (γn yn ) → h2 (y∞′ ). Next we prove cocompactness. Let (yn )n be a sequence in Ω2 . For every n ∈ N we consider zn ∈ Ω1 as above: ξ1 (zn ) = ξ2 (yn ), η1 (zn ) = f∞ (η2 (yn )), and h1 (zn ) = h2 (yn ). As the action is cocompact in Ω1 , there exists a sequence γn in Γ such that γn zn converges, and all we need to prove is that |h2 (γn yn )| is bounded. This is a consequence of the inequality |h2 (γn yn ) − h1 (γn zn )| ≤ 6K (by Lemma 6.20) and that h1 (γn zn ) converges. □ Now we consider the action on the whole X1 × X2 ∪ Ω2 . We require the following lemma: Lemma 6.22. Let (xn , yn )n be a diverging sequence in X1 × X1 . The accu- mulation set of (xn , yn )n is contained in Ω1 if and only if the accumulation set of (xn , f (yn ))n is contained in Ω2 . Proof. First assume that (xn , yn )n converges to a point in Ω1 . Namely xn → ξ ∈ ∂∞ X1 , yn → η ̸= ξ ∈ ∂∞ X1 and |d1 (xn , o) − d1 (yn , o)| is bounded. Thus, as xn → ξ and f (yn ) → f∞ (η) ̸= f∞ (ξ), the assertion follows from the estimate |d1 (xn , o) − d2 (f (yn ), fn (o))| ≤ |d1 (xn , o) − d1 (yn , o)| + |d1 (yn , o) − d2 (f (yn ), fn (o))|, that is bounded because f is K-almost isometry. ACTIONS ON PRODUCTS OF CAT(−1) SPACES 25 For the converse, assuming that |d1 (xn , o) − d2 (yn , f (o))| is bounded, we write: |d1 (xn , o) − d1 (f −1 (yn ), o)| ≤ |d1 (xn , o) − d2 (yn , f (o))| + |d2 (yn , f (o)) − d1 (f −1 (yn ), o)|, that is bounded because: |d2 (yn , f (o)) − d1 (f −1 (yn ), o)| ≤ |d2 (yn , f (o)) − d1 (f −1 (yn ), f −1 (f (o)))| + |d1 (f −1 (yn ), f −1 (f (o))) − d1 (f −1 (yn ), o)| ≤ |d2 (yn , f (o)) − d1 (f −1 (yn ), f −1 (f (o)))| + d1 (f −1 (f (o)), o) ≤ 2K. □ Theorem 6.23. The action of Γ on X1 × X2 ∪ Ω2 is properly discontinuous and cocompact. Proof. For proper discontinuity we will prove that no two points in X1 ×X2 ∪ Ω2 are dynamically related, as in the proof of Proposition 6.21. Since the ac- tion is properly discontinuous on both X1 × X2 and Ω2 , it is enough to check that if (xn , yn ) is a sequence in X1 × X2 that converges to a point z ∈ Ω2 , then there is no divergent sequence (γn )n ⊂ Γ such that (ρ1 (γn )xn , ρ2 (γn )yn ) accumulates in X1 × X2 ∪ Ω2 . By contradiction, assume that such sequences exist. If (ρ1 (γn )xn , ρ2 (γn )yn ) converges to a point (x, y) ∈ X1 × X2 , then dmax ((ρ1 (γn−1 )(x), ρ2 (γn−1 )(y)), (xn , yn )) is uniformly bounded and (ρ1 (γn−1 )(x), ρ2 (γn−1 )(y))n converges to the same point as (xn , yn )n . Hence z ∈ Ω2 is the accumulation point of an orbit and we get a contradiction with Remark 6.18. Therefore, we assume that (ρ1 (γn )xn , ρ2 (γn )yn ) accumulates in Ω2 . 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