Relatively dominated representations from eigenvalue gaps and limit maps
aa r X i v : . [ m a t h . G R ] F e b RELATIVELY DOMINATED REPRESENTATIONS FROMEIGENVALUE GAPS AND LIMIT MAPS
FENG ZHU
Abstract.
Relatively dominated representations give a common generaliza-tion of geometrically finiteness in rank one on the one hand, and the Anosovcondition which serves as a higher-rank analogue of convex cocompactness onthe other. This note proves three results about these representations.Firstly, we remove the quadratic gaps assumption involved in the originaldefinition. Secondly, we give a characterization using eigenvalue gaps, provid-ing a relative analogue of a result of Kassel–Potrie for Anosov representations.Thirdly, we formulate characterizations in terms of singular value or eigenvaluegaps combined with limit maps, in the spirit of Gu´eritaud–Guichard–Kassel–Wienhard for Anosov representations, and use them to show that inclusionrepresentations of certain groups playing weak ping-pong and positive repre-sentations in the sense of Fock–Goncharov are relatively dominated. Introduction
Relatively dominated representations were introduced in [Zhu19] and provide acommon generalization of geometric finiteness in rank-one semisimple Lie groupsand the Anosov condition in more general semisimple Lie groups. They are relatedto earlier common generalizations studied by Kapovich and Leeb in [KL18]. Theserepresentations furnish a class of discrete, faithful relatively hyperbolic subgroupsof semisimple Lie groups which are quasi-isometrically embedded modulo controlleddistortion along their peripheral subgroups.The definition of these representations is given in terms of singular value gaps,which may be interpreted in terms of the geometry of the associated symmetricspaces as distances from singular flats of specified type. The corresponding char-acterization of Anosov representations was given first by Kapovich–Leeb–Porti in[KLP17] under the name of URU subgroups, and subsequently reformulated, inlanguage more closely resembling that used here, by Bochi–Potrie–Sambarino in[BPS19].The key defining condition asserts that the singular value gap σ σ ( ρ ( γ )) growsuniformly linearly in a notion of word-length | γ | c that has been modified to takeinto account the distortion along the peripheral subgroups.The definition also involves additional technical conditions to control the imagesof the peripheral subgroups. In the first part of this note, we remove one of thosetechnical conditions by showing that it follows from the other parts of the definition.We refer the reader to §
3, and specifically Proposition 3.5, for the full statementand proof; here we present it slightly summarised as follows: F. Zhu
Proposition 1.1.
Let Γ be a finitely-generated group, and fix constants C, c > .Suppose we have a representation ρ : Γ → SL( d, R ) such that C − log σ σ ( ρ ( γ )) − c ≤ | γ | c ≤ C log σ σ ( ρ ( γ )) + c for all γ ∈ Γ .Then, given constants ¯ υ, ¯ υ > , there exists constants C, µ > such that for anybi-infinite sequence of elements in Γ ( γ n ) n ∈ Z satisfying(i) γ = id , and(ii) ¯ υ − | n | − ¯ υ ≤ | γ n | c ≤ ¯ υ | n | + ¯ υ for all n ,and any k ∈ Z , d ( U ( ρ ( γ k − · · · γ k − n )) , U ( ρ ( γ k − · · · γ k − n − ))) < Ce − µn for all n > . Here U ( B ) denotes the image of the 1-dimensional subspace of R d most ex-panded by B . This proposition allows us to obtain uniform convergence towardslimit points; the exponential convergence seen here is reminiscent of hyperbolicdynamics, and is straightforward to obtain in the non-relative case.In the proof of Proposition 3.5 we will find it useful to adopt elements of the pointof view of Kapovich–Leeb–Porti, which emphasizes the geometry of the symmetricspace and the related geometry of its boundary and associated flag spaces.More recently, Kassel–Potrie [KP20] have given a characterization of Anosovrepresentations in terms of eigenvalue gaps, which may be interpreted as asymptoticversions of singular value gaps, i.e. distance to the Weyl chamber walls at infinity.In the second part of this note, we give an analogous characterization of relativelydominated representations: Theorem 1.2 (Theorem 4.1) . Let Γ be hyperbolic relative to P . A representation ρ : Γ → SL( d, R ) is P -dominated relative to P if and only if the following fourconditions hold: • (weak eigenvalue gaps) there exist constants ¯ C, ¯ µ > such that λ λ ( ρ ( γ )) ≥ ¯ Ce ¯ µ | γ | c, ∞ for all γ ∈ Γ , • (upper eigenvalue gaps) there exist constants ¯ C, ¯ µ > such that λ λ d ( ρ ( γ )) ≤ ¯ Ce ¯ µ | γ | c, ∞ for all γ ∈ Γ , • (unique limits) for each P ∈ P , there exists ξ ρ ( P ) ∈ P ( R d ) and ξ ∗ ρ ( P ) ∈ Gr d − ( R d ) such that for every sequence ( η n ) ⊂ P with η n → ∞ , we have lim n →∞ U ( ρ ( η n )) = ξ ρ ( P ) and lim n →∞ U d − ( ρ ( η n )) = ξ ∗ ρ ( P ) . • (uniform transversality) for every P, P ′ ∈ P and γ ∈ Γ , ξ ( P ) = ξ ( γP ′ γ − ) .Moreover, for every ¯ υ, ¯ υ > , there exists δ > such that for all P, P ′ ∈ P and g, h ∈ Γ such that there exists a bi-infinite (¯ υ, ¯ υ ) -metric quasigeodesicpath ηghη ′ where η ′ is in P ′ and η is in P , we have sin ∠ ( g − ξ ( P ) , h ξ ∗ ( P ′ )) > δ . (See Definition 2.8 for the precise definition of a metric quasigeodesic path.) elatively dominated representations: gaps and maps Here | γ | c, ∞ is a stable version of the modified word-length | γ | c ; we refer thereader to § Theorem 1.3 (Theorem 5.1 and Corollary 5.2) . Given (Γ , P ) a relatively hyperbolicgroup, a representation ρ : Γ → SL( d, R ) is P -dominated relative to P if and onlyif • there exist continuous, ρ -equivariant, transverse limit maps ξ ρ : ∂ (Γ , P ) → P ( R d ) and ξ ∗ ρ : ∂ (Γ , P ) → P ( R d ∗ ) . and one of the following (equivalent) sets of conditions holds: (D − ) there exist constants ¯ C, ¯ µ > such that for all γ ∈ Γ , and σ σ ( ρ ( γ )) ≥ ¯ Ce ¯ µ | γ | c , (D+) there exist constants C , µ > such that σ ( ρ ( η )) ≤ C e µ | η | c for everyperipheral element η ∈ S P ; or • (weak eigenvalue gaps) there exist constants ¯ C, ¯ µ > such that λ λ ( ρ ( γ )) ≥ ¯ Ce ¯ µ | γ | c, ∞ for all γ ∈ Γ , and • (upper eigenvalue gaps) there exist constants ¯ C, ¯ µ > such that λ λ d ( ρ ( γ )) ≤ ¯ Ce ¯ µ | γ | c, ∞ for all γ ∈ Γ . As an application of this, we show that certain free groups which contain unipo-tent generators and which play weak ping-pong in projective space, including im-ages of positive representations in the sense of [FG06], are relatively P -dominated(Examples 5.3 and 5.6 below). Organization. § § § § F. Zhu
Acknowledgements.
The author thanks Max Riestenberg for helpful conversa-tions about the Kapovich–Leeb–Porti approach to Anosov representations, KostasTsouvalas for stimulating comments, and Fran¸cois Gu´eritaud and Jean-PhilippeBurelle for helpful discussions related to ping-pong and positive representations.The author acknowledges support from ISF grant 871/17.This research was conducted during the covid-19 pandemic. The author extendshis heartfelt gratitude to all those—friends, family, mentors, funding agencies—whose combined efforts have given him safe harbor in these tumultuous times.2.
Preliminaries
Relatively hyperbolic groups and cusped spaces.
Relative hyperbolicityis a group-theoretic notion of non-positive curvature inspired by the geometry ofcusped hyperbolic manifolds and free products.Consider a finite-volume cusped hyperbolic manifold with an open neighborhoodof each cusp removed: call the resulting truncated manifold M . The universalcover ˜ M of such a M is hyperbolic space with a countable set of horoballs removed.The universal cover ˜ M is not Gromov-hyperbolic; distances along horospheres thatbound removed horoballs are distorted. If we glue the removed horoballs back into the universal cover, however, the resulting space will again be hyperbolic space.Gromov generalized this in [Gro87, § P if (Γ , P ) admits a cusp-uniform action on a hyperbolic metric space X , meaning there exists some system( H P ) P ∈P of disjoint horoballs of X , each preserved by a subgroup P ∈ P , such thatthe Γ acts on X discretely and isometrically, and the Γ action on X r S P H P iscocompact.The space X is sometimes called a Gromov model for (Γ , P ). There isin generalno canonical Gromov model for a given relatively hyperbolic group, but there aresystematic constructions one can give, one of which we describe here. The descrip-tion below, as well as the material in the next section § §
2] and is based on prior literature, in particular [GM08]; it is included here forcompleteness.
Definition 2.1 ([GM08, Definition 3.1]) . Given a subgraph Λ of the Cayley graph Cay(Γ , S ) , the combinatorial horoball based on Λ , denoted H = H (Λ) , is the1-complex formed as follows: • the vertex set H (0) is given by Λ (0) × Z ≥ • the edge set H (1) consists of the following two types of edges:(1) If k ≥ and v and w ∈ Λ (0) are such that < d Λ ( v, w ) ≤ k , thenthere is a (“horizontal”) edge connecting ( v, k ) to ( w, k ) (2) If k ≥ and v ∈ Λ (0) , there is a (“vertical”) edge joining ( v, k ) to ( v, k + 1) . H is metrized by assigning length 1 to all edges. Next let P be a finite collection of finitely-generated subgroups of Γ, and suppose S is a compatible generating set , i.e. for each P ∈ P , S ∩ P generates P . Groves-Manning combinatorial horoballs are actually defined as 2-complexes; the definitionhere is really of a 1-skeleton of a Groves-Manning horoball. For metric purposes only the 1-skeletonmatters. elatively dominated representations: gaps and maps Definition 2.2 ([GM08, Definition 3.12]) . Given Γ , P , S as above, the cuspedspace X (Γ , P , S ) is the simplicial metric graph Cay(Γ , S ) ∪ [ H ( γP ) where the union is taken over all left cosets of elements of P , i.e. over P ∈ P and(for each P ) γP in a collection of representatives for left cosets of P .Here the induced subgraph of H ( γP ) on the γP × { } vertices is identified with(the induced subgraph of ) γP ⊂ Cay(Γ , S ) in the natural way. Definition 2.3. Γ is hyperbolic relative to P if and only if the cusped space X (Γ , P , S ) is δ -hyperbolic (for any compatible generating set S ; the hyperbolicityconstant δ may depend on S .)We will also call (Γ , P ) a relatively hyperbolic structure . We remark that for a fixed relatively hyperbolic structure (Γ , P ), any two cuspedspaces, corresponding to different compatible generating sets S , are quasi-isometric[Gro13, Corollary 6.7]: in particular, the notion above is well-defined independentof the choice of generating set S . There is a natural action of Γ on the cuspedspace X = X (Γ , P , S ); with respect to this action, the quasi-isometry between twocusped spaces X (Γ , P , S i ) ( i = 1 ,
2) is Γ-equivariant.In particular, this gives us a notion of a boundary associated to the data of arelatively hyperbolic group Γ and its peripheral subgroups P : Definition 2.4.
For Γ hyperbolic relative to P , the Bowditch boundary ∂ (Γ , P ) is defined as the Gromov boundary ∂ ∞ X of any cusped space X = X (Γ , P , S ) . By the remarks above, this is well-defined up to homeomorphism, independentof the choice of compatible generating set S [Bow12, § P and S as above, for γ, γ ′ ∈ Γ, d ( γ, γ ′ ) willdenote the distance between γ and γ ′ in the Cayley graph with the word metric,and | γ | := d (id , γ ) denotes word length in this metric. Similarly, d c ( γ, γ ′ ) denotesdistance in the corresponding cusped space and | γ | c := d c (id , γ ) denotes cuspedword-length.2.2. Geodesics in the cusped space.
Let Γ be a finitely-generated group, P bea malnormal finite collection of finitely-generated subgroups, and let S = S − be acompatible finite generating set as above. Let X = X (Γ , P , S ) be the cusped space,and Cay(Γ) = Cay(Γ , S ) the Cayley graph. Here we collect some technical resultsabout geodesics in these spaces that will be useful below. Lemma 2.5 ([GM08, Lemma 3.10]) . Let H (Γ) be a combinatorial horoball. Sup-pose that x, y ∈ H (Γ) are distinct vertices. Then there is a geodesic γ ( x, y ) = γ ( y, x ) between x and y which consists of at most two vertical segments and a single hori-zontal segment of length at most 3. We will call any such geodesic a preferred geodesic .Given a path γ : I → Cay(Γ) in the Cayley graph such that γ ( I ∩ Z ) ⊂ Γ, we canconsider γ as a relative path ( γ, H ), where H is a subset of I consisting of a disjointunion of finitely many subintervals H , . . . , H n occurring in this order along I , suchthat each η i := γ | H i is a maximal subpath lying in a closed combinatorial horoball B i , and γ | I r H contains no edges of Cay(Γ) labelled by a peripheral generator. F. Zhu
Similarly, a path ˆ γ : ˆ I → X in the cusped space with endpoints in Cay(Γ) ⊂ X may be considered as a relative path (ˆ γ, ˆ H ), where ˆ H = ` ni =1 ˆ H i , ˆ H , . . . , ˆ H n occurin this order along ˆ I , each ˆ η i := ˆ γ | ˆ H i is a maximal subpath in a closed combinatorialhoroball B i , and ˆ γ | ˆ I r ˆ H lies inside the Cayley graph. Below, we will consider onlygeodesics and quasigeodesic paths ˆ γ : ˆ I → X where all of the ˆ η i are preferredgeodesics (in the sense of Lemma 2.5.)We will refer to the η i and ˆ η i as peripheral excursions . We remark that the η i , or any other subpath of γ in the Cayley graph, may be considered as a wordand hence a group element in Γ; this will be used without further comment below.Given a path ˆ γ : ˆ I → X whose peripheral excursions are all preferred geodesics,we may replace each excursion ˆ η i = ˆ γ | ˆ H i into a combinatorial horoball with a ge-odesic path (or, more precisely, a path with geodesic image) η i = π ◦ ˆ η i in theCayley (sub)graph of the corresponding peripheral subgroup connecting the sameendpoints, by omitting the vertical segments of the preferred geodesic ˆ η i and replac-ing the horizontal segment with the corresponding segment at level 0, i.e. in theCayley graph. We call this the “project” operation, since it involves “projecting”paths inside combinatorial horoballs onto the boundaries of those horoballs. Thisproduces a path γ = π ◦ ˆ γ : ˆ I → Cay(Γ).Below, given any path α in the Cayley graph with endpoints g, h ∈ Γ, or anypath ˆ α in the cusped space with endpoints in g, h ∈ X , we write ℓ ( α ) to denote d ( g, h ), i.e. distance measured according to the word metric in Cay(Γ), and ℓ c (ˆ α )to denote d c ( g, h ), where d c denotes distance in the cusped space.We have the following biLipschitz equivalence between cusped distances andsuitably-modified distances in the Cayley graph: Proposition 2.6 ([Zhu19, Proposition 2.12]) . Given a geodesic ˆ γ : ˆ J → X with endpoints in Cay(Γ) ⊂ X and whose peripheral excursions are all preferredgeodesics, let γ = π ◦ ˆ γ : ˆ J → Cay(Γ) be its projected image.Given any subinterval [ a, b ] ⊂ ˆ J , consider the subpath γ | [ a,b ] as a relative path ( γ | [ a,b ] , H ) where H = ( H , . . . , H n ) , and write η i := γ | H i ; then we have ≤ d c ( γ ( a ) , γ ( b )) ℓ ( γ | [ a,b ] ) − P ni =1 ℓ ( η i ) + P ni =1 ˆ ℓ ( η i ) ≤ < where ˆ ℓ ( η i ) := max { log( ℓ ( η i )) , } . Below we will occasionally find it useful to consider paths in Cay(Γ) that “behavemetrically like quasi-geodesics in the relative Cayley graph”, in the following sense:
Definition 2.7.
Given any path γ : I → Cay(Γ) such that I has integer endpointsand γ ( I ∩ Z ) ⊂ Γ , define the depth δ ( n ) = δ γ ( n ) of a point γ ( n ) (for any n ∈ I ∩ Z )as(a) the smallest integer d ≥ such that at least one of γ ( n − d ) , γ ( n + d ) is well-defined (i.e. { n − d, n + d } ∩ I = ∅ ) and not in the same peripheral coset as γ ( n ) , or (b) if no such integer exists, min { sup I − n, n − inf I } . Definition 2.8.
Given constants ¯ υ, ¯ υ > , an (¯ υ, ¯ υ ) -metric quasigeodesic path is a path γ : I → Cay(Γ) with γ ( I ∩ Z ) ⊂ Γ such that for all integers m, n ∈ I , As a parametrized path this has constant image on the subintervals of ˆ H i corresponding tothe vertical segments, and travels along the projected horizontal segment at constant speed. elatively dominated representations: gaps and maps (i) | γ ( n ) − γ ( m ) | c ≥ ¯ υ − | m − n | − ¯ υ ,(ii) | γ ( n ) − γ ( m ) | c ≤ ¯ υ ( | m − n | + min { δ ( m ) , δ ( n ) } ) + ¯ υ , and(iii) if γ ( n ) − γ ( n +1) ∈ P for some P ∈ P , we have γ ( n ) − γ ( n +1) = p n, · · · p n,ℓ ( n ) where each p n,i is a peripheral generator of P , and δ ( n ) − ≤ ℓ ( n ) := | γ ( n ) − γ ( n + 1) | ≤ δ ( n )+1 . The terminology comes from the following fact: given a geodesic segment ˆ γ inthe cusped space with endpoints in Cay(Γ), we can project the entire segment to theCayley graph and reparametrize the projected image to be a metric quasigeodesicpath—the idea being that in such a reparametrization, the increments correspond,approximately, to linear increments in cusped distance: see the discussion in [Zhu19, § Floyd boundaries.
Let Γ be a finitely generating group, and S a finite gen-erating set giving a word metric | · | .A Floyd boundary ∂ f Γ for Γ is a boundary for Γ meant to generalize the idealboundary of a Kleinian group. Its construction uses the auxiliary data of a
Floydfunction , which is a function f : N → R > satisfying(i) P ∞ n =1 f ( n ) < ∞ , and(ii) there exists m > m ≤ f ( k +1) f ( k ) ≤ k ∈ N .Given such a function, there exists a metric d f on Γ defined by setting d f ( g, h ) = f (max {| g | , | h |} ) if g, h are adjacent vertices in Cay(Γ , S ), and considering the re-sulting path metric. Then the Floyd boundary ∂ f Γ with respect to f is givenby ∂ f Γ := ¯Γ r Γwhere ¯Γ is the metric completion of Γ with respect to the metric d f .Below, the Floyd boundary, in particular the ability of the Floyd function toserve as a sort of “distance to infinity”, will be useful as a tool in the proof ofTheorem 4.1. It may be possible, with more work, to replace the role of the Floydboundary in that proof with the Bowditch boundary.The Floyd boundary ∂ f Γ is called non-trivial if it has at least three points.Gerasimov and Potyagailo have studied Floyd boundaries of relatively hyperbolicgroups:
Theorem 2.9 ([Ger12], [GP13]) . Suppose we have a non-elementary relativelyhyperbolic group Γ which is hyperbolic relative to P .Then there exists a Floyd function f such that ∂ f Γ is non-trivial, and moreover(a) there exists a continuous equivariant map F : ∂ f G → ∂ (Γ , P ) , such that(b) for any parabolic point p ∈ ∂ (Γ , P ) , we have F − ( p ) = ∂ f (Stab Γ p ) , and if thereexist a = b such that F ( a ) = F ( b ) = p , then p is parabolic. Remark 2.10.
It is an open question whether every group with a non-trivial Floydboundary is relatively hyperbolic—see e.g. [Lev20].For more details, including justifications for some of the assertion above, we referthe reader to [Flo80] and [Kar03].
F. Zhu
Gromov products and translation lengths in hyperbolic spaces.
Wecollect and state here, for the reader’s convenience, assorted facts about Gromovproducts and translation lengths in Gromov-hyperbolic spaces that we use below,in particular in and around the statement and proof of Theorem 4.1.Given (
X, d ) a proper geodesic metric space, x ∈ X a fixed basepoint, and γ anisometry of X , we define the translation length of γ as ℓ X ( γ ) := inf x ∈ X d ( γx, x )and the stable translation length of γ as | γ | X, ∞ := lim n →∞ d ( γ n x , x ) n . When X is δ -hyperbolic space, these two quantities are coarsely equivalent: Proposition 2.11 ([CDP90, Chapitre 10, Proposition 6.4]) . For ( X, d ) a δ -hyperbolicspace, the quantities ℓ X ( γ ) and | γ | X, ∞ defined above satisfy ℓ X ( γ ) − δ ≤ | γ | X, ∞ ≤ ℓ X ( γ ) . The
Gromov product with respect to x is the function h· , ·i x : X × X → R defined by h x, y i x := 12 ( d ( x, x ) + d ( y, x ) − d ( x, y )) . There is a relation between the Gromov product, the stable translation length | γ | X, ∞ , and the quantity | γ | X = d ( γx , x ), given by Lemma 2.12.
Given ( X, d ) a proper geodesic metric space, x ∈ X a basepoint,and γ an isometry of X , we can find a sequence of integers ( m i ) i ∈ N i →∞ h γ m i , γ − i x ≥ | γ | X − | γ | X, ∞ . Proof.
By the definition of the stable translation length, we can find a sequence( m i ) i ∈ N such that lim i →∞ (cid:0) | γ m i +1 | X − | γ m i | X (cid:1) ≤ | γ | X, ∞ . By the definition of the Gromov product,2 h γ m i , γ − i x := | γ m i | X + d ( γ − x , x ) − d ( γ m i x , γ − x ) . Since γ acts isometrically on X , d ( γ m i x , γ − x ) = | γ m − i +1 | X and d ( γ − x , x ) = | γ | X . Then we have2 h γ m i , γ − i x = | γ m i | X + | γ | X − | γ m i +1 | X ≤ | γ | X − | γ | X, ∞ as desired. (cid:3) Singular value decompositions.
We collect here facts about singular valuesand Cartan decomposition in SL( d, R ). The defining conditions for our representa-tions will be phrased, in the first instance, in terms of these, and more generally theywill be helpful for understanding the geometry associated to our representations.Given a matrix g ∈ GL( d, R ), let σ i ( g ) (for 1 ≤ i ≤ d ) denote its i th singularvalue, and write U i ( g ) to denote the span of the i largest axes in the image of theunit sphere in R d under g , and S i ( g ) := U i ( g − ). Note U i ( g ) is well-defined if andonly if we have a singular-value gap σ i ( g ) > σ i +1 ( g ).More algebraically, given g ∈ GL( d, R ), we may write g = KAL , where K and L are orthogonal matrices and A is a diagonal matrix with nonincreasing positive elatively dominated representations: gaps and maps entries down the diagonal. A is uniquely determined, and we may define σ i ( g ) = A ii ; U i ( g ) is given by the span of the first i columns of K .For g ∈ SL( d, R ), this singular-value decomposition is a concrete manifestation ofa more general Lie-theoretic object, a (particular choice of) Cartan decompositionSL( d, R ) = SO( d ) · exp( a + ) · SO( d ), where SO( d ) is the maximal compact subgroupof SL( d, R ), and a + is a positive Weyl chamber.We recall that there is an adjoint action Ad of SL( d, R ) on sl ( d, R ).We will occasionally write (given g = KAL as above) a ( g ) := (log A , . . . , log A dd ) = (log σ ( g ) , . . . , log σ d ( g ));we note that the norm k a ( g ) k = p (log σ ( g )) + · · · + (log σ d ( g )) is equal to thedistance d ( o, g · o ) in the associated symmetric space SL( d, R ) / SO( d ) (see e.g. for-mula (7.3) in [BPS19].)3. Relatively dominated representations
Definition 3.1 ([Zhu19, § . Let Γ be a finitely-generated torsion-free group whichis hyperbolic relative to a collection P of proper infinite subgroups.Let S be a compatible generating set, and let X = X (Γ , P , S ) be the correspondingcusped space (see Definitions 2.1 and 2.2 above.) As above, let d c denote the metricon X , and | · | c := d c (id , · ) denote the cusped word-length.Fix constants ¯ C, ¯ µ > . A representation ρ : Γ → GL( d, R ) is P -dominatedrelative to P with lower domination constants (¯ C, ¯ µ ) , if it satisfies (D − ) for all γ ∈ Γ , σ σ ( ρ ( γ )) ≥ ¯ Ce ¯ µ | γ | c ,and the images of peripheral subgroups under ρ are well-behaved, meaning (D+) there exist constants ¯ C, ¯ µ > such that σ σ d ( ρ ( η )) ≤ ¯ Ce ¯ µ | η | c for every γ ∈ Γ ; • (unique limits) for each P ∈ P , there exists ξ ρ ( P ) ∈ P ( R d ) and ξ ∗ ρ ( P ) ∈ Gr d − ( R d ) such that for every sequence ( η n ) ⊂ P with η n → ∞ , we have lim n →∞ U ( ρ ( η n )) = ξ ρ ( P ) and lim n →∞ U d − ( ρ ( η n )) = ξ ∗ ρ ( P ) ; • (uniform transversality) for every P, P ′ ∈ P and γ ∈ Γ , ξ ( P ) = ξ ( γP ′ γ − ) .Moreover, for every ¯ υ, ¯ υ > , there exists δ > such that for all P, P ′ ∈ P and g, h ∈ Γ such that there exists a bi-infinite (¯ υ, ¯ υ ) -metric quasigeodesicpath ηghη ′ where η ′ is in P ′ and η is in P , we have sin ∠ ( g − ξ ( P ) , h ξ ∗ ( P ′ )) > δ . Remark 3.2.
Since Γ is finitely generated, so are its peripheral subgroups, by[DGO17, Proposition 4.28 and Corollary 4.32].
Remark 3.3.
It is also possible to formulate the definition without assuming rela-tive hyperbolicity, if one imposes additional hypotheses on the peripheral subgroups P ; it is then possible to show that any group admitting such a representation mustbe hyperbolic relative to P : see [Zhu19] for details.The definition which originally appeared in [Zhu19] also had an additional “qua-dratic gaps” hypothesis, as part of the definition of the peripheral subgroups havingwell-behaved images. The only input of this assumption into the subsequent resultsthere was in [Zhu19, Lemma 5.4]; the next proposition obtains the conclusion ofthat lemma from the other hypotheses (not including relative hyperbolicity), with-out using the quadratic gaps hypothesis. F. Zhu
Definition 3.4.
Let α : Z → Cay(Γ) be a bi-infinite path with α ( Z ) ⊂ Γ .We define the sequence x γ = ( . . . A a − , . . . , A − , A , . . . , A b − , . . . ):= ( . . . , ρ ( α ( a ) − α ( a − , . . . , ρ ( α (0) − α ( − , ρ ( α (1) − α (0)) , . . . , ρ ( α ( b ) − α ( b − , . . . ) and call this the matrix sequence associated to α . Proposition 3.5.
Given a representation ρ : Γ → SL( d, R ) satisfying (D ± ), andgiven ¯ υ, ¯ υ > , there exist constants C ≥ and µ > , depending only on therepresentation ρ and ¯ υ, ¯ υ , such that for any matrix sequence x = x γ associated toa bi-infinite (¯ υ, ¯ υ ) -metric quasigeodesic path γ with γ (0) = id , d ( U ( A k − · · · A k − n ) , U ( A k − · · · A k − ( n +1) )) ≤ Ce − nµ d ( S d − ( A k + n − · · · A k ) , S d − ( A k + n · · · A k )) ≤ Ce − nµ . Proof.
Given (D ± ), there exists C r , c r > σ σ ( ρ ( γ )) ≥ C r log σ σ d ( ρ ( γ )) − c r for all γ ∈ Γ. (Specifically, we can take C r = ¯ µ/ ¯ µ and c r = (¯ µ/ ¯ µ ) log ¯ C − log ¯ C .)In the language of Kapovich–Leeb–Porti—see [KLP17], or [KL18] for the relativecase; we adapt the relevant parts of this language and framework here— ρ (Γ) is auniformly regular subgroup of SL( d, R ).Given fixed constants C r , c r , a matrix B ∈ SL( d, R ) will be called ( P , C r , c r ) -regular if it satisfies log σ σ ( B ) ≥ C r log σ σ d ( B ) − c r . The set of all limit points inthe visual boundary of the symmetric space of sequences ( B n · o ), where o variesover all possible basepoints in the symmetric space and ( B n ) over all divergent( P , C r , c r )-regular sequences with all c r >
0, will be called the ( P , C r ) -regularideal points .Roughly speaking, geodesics converging to such regular ideal points have as manyhyperbolic directions as possible in the symmetric space, and thus flag convergencealong these geodesics should occur exponentially quickly, just as in the hyperboliccase. This intuition can be made precise with more work, as follows:For fixed C r , the set of ( P , C r )-regular ideal points is compact. There is asmooth map π from the set of ( P , C r )-regular ideal points to P ( R d ) given by takinglim n B n · o to lim n →∞ U ( B n ). π is Lipschitz by the compactness of the domain,with Lipschitz constant depending only on the regularity constant C r and the choiceof basepoint o implicit in the measurement of the singular values ([KLP17, § § § ρ ( γ ) is ( P , C r , c r )-regular for any γ ∈ Γ, given the Cartandecomposition ρ ( γ ) = K γ · exp( a ( ρ ( γ ))) · L γ , we haveΞ ρ ( γ ) = π (cid:16) lim n →∞ K γ · exp( na ( ρ ( γ ))) · L γ · o (cid:17) . Thus, given any sequence ( γ n ) ⊂ Γ, we have d (Ξ ρ ( γ n ) , Ξ ρ ( γ m )) ≤ C Lip · sin ∠ (Ad( K γ n ) a ( ρ ( γ n )) , Ad( K γ m ) a ( ρ ( γ m ))) . Now observe that, if x = x γ = ( A n ) n ∈ N is a matrix associated to a bi-infinite(¯ υ, ¯ υ )-metric quasigeodesic path γ with γ (0) = id, then A k − . . . A k − n = ρ ( γ ( k ) − γ ( k − elatively dominated representations: gaps and maps n )). We write ρ ( γ ( k ) − γ ( k − n )) = K k,n · exp( a ( k, n )) · L k,n to denote the parts ofthe Cartan decomposition.By ( P , C r , c r )-regularity and the higher-rank Morse lemma [KLP18a, Theorem1.1], the limitlim n →∞ U ( A k − · · · A k − n ) = lim n →∞ K k,n h e i = lim n →∞ Ad( K k,n ) a ( k, n )exists, and we have a bound C a on the distance from A k − · · · A k − n · o to a nearestpoint on any ( P , C r )-regular ray ( B n · o ) starting at o such that lim n →∞ U ( B n ) =lim n K k,n h e i (below, we refer to this point as π lim A k − · · · A k − n · o ), where C a depends only on C r , c r and ¯ υ, ¯ υ .Then, by [Rie21, Lemma 4.8] applied with p = o our basepoint, α = C r , τ amodel Weyl chamber corresponding to the first simple root / singular value gap, q = A k − · · · A k − n · o , r = π lim q , 2 l = k a ( k, n ) k ≥ ¯ υ − n − ¯ υ and D = C a , we havesin ∠ (cid:16) Ad( K k,n ) a ( k, n ) , lim n K k,n h e i (cid:17) = sin ∠ (cid:18)
12 Ad( K k,n ) a ( k, n ) , lim n K k,n h e i (cid:19) ≤ d ( q/ , π lim q/ d ( o, π lim q/ ≤ C a e C a / √ d +¯ υ/ e − ( C r / υ ) n d ( o, π lim q/ ≤ C a e C a / √ d +¯ υ/ e − ( C r / υ ) n once n is sufficiently large, depending only on the dimension d , our constants C r , C a and choice of basepoint o ; here q/ oq , which can be writtenas K k,n · exp (cid:18) a ( k, n ) (cid:19) · L k,n · o. Hence we can find ˆ C ≥ C a e C a / √ d +¯ υ/ such thatsin ∠ (cid:16) Ad( K k,n ) a ( k, n ) , lim n K k,n h e i (cid:17) ≤ ˆ Ce − ( C r / υ ) n for all n , and so d ( U ( A k − · · · A k − n ) , U ( A k − · · · A k − n − )) is bounded above by C Lip sin ∠ (Ad( K k,n ) a ( k, n ) , Ad( K k,n +1 ) a ( k, n + 1)) ≤ C Lip ˆ C (cid:16) e − C r / υ (cid:17) e − ( C r / υ ) n . This gives us the desired bound with µ = C r ¯ υ − = ¯ µ (¯ µ ¯ υ ) − and C = C Lip ˆ C (1 + e − µ ).We can obtain the analogous bound for d ( S d − ( A k + n − · · · A k ) , S d − ( A k + n · · · A k ))by arguing similarly, or working with the dual representation—for the details of thispart we refer the interested reader to the end of the proof of [Zhu19, Lemma 5.4]. (cid:3) A characterisation using eigenvalue gaps
Suppose Γ is hyperbolic relative to P . We have, as above, the cusped space X = X (Γ , P ), which is a δ -hyperbolic space on which Γ acts isometrically and For readers more acquainted with the language of Kapovich–Leeb–Porti: this is the distanceto the Weyl cone over the C r -regular open star of lim n K k,n h e i . F. Zhu properly. We define | · | c, ∞ to be the stable translation length on this space, i.e. | γ | c, ∞ := lim n →∞ | γ n | c n where | · | c := d X (id , · ) as above.We remark that by Proposition 2.11 the eigenvalue gap conditions below maybe equivalently formulated in terms of the translation length ℓ X ( γ ).Given A ∈ GL( d, R ), let λ i ( A ) denote the magnitude of the i th largest eigenvalueof A . We will prove Theorem 4.1.
Let Γ be hyperbolic relative to P . A representation ρ : Γ → SL( d, R ) is P -dominated relative to P if and only if the following four conditions hold: • (weak eigenvalue gaps) there exist constants ¯ C, ¯ µ > such that λ λ ( ρ ( γ )) ≥ ¯ Ce ¯ µ | γ | c, ∞ for all γ ∈ Γ , • (upper eigenvalue gaps) there exist constants ¯ C, ¯ µ > such that λ λ d ( ρ ( γ )) ≤ ¯ Ce ¯ µ | γ | c, ∞ for all γ ∈ Γ , • (unique limits) as in Definition 3.1 and • (uniform transversality) as in Definition 3.1, Remark 4.2.
We remark that the upper eigenvalue gaps condition really is equiva-lent to requiring if η is a peripheral element (so | η | c, ∞ = 0), then all the eigenvalueof ρ ( η ) have absolute value 1. If γ is such that γ | c, ∞ = | γ | ∞ , then the conditionalways holds because Γ is finitely-generated; more generally we have λ ( ρ ( γη )) ≤ λ ( ρ ( γ )) λ ( ρ ( η )) , and λ d ( ρ ( γη )) = λ ( ρ ( η − γ − )) − ≥ λ d ( ρ ( γ )) λ d ( ρ ( η )) , and we can use these to piece together the condition on non-peripheral and periph-eral parts of the word for γ . Proof.
It suffices to show that the weak and upper eigenvalue gap conditions areequivalent to the (D ± ) conditions.We recall the identity log λ i ( A ) = lim n →∞ log σ ( A n ) n . Given (D − ), we have(log λ − log λ )( ρ ( γ )) = lim n →∞ n (log σ − log σ )( ρ ( γ n )) ≥ lim n →∞ n (log ¯ C + ¯ µ | γ n | c ) = ¯ µ | γ | c, ∞ and so λ λ ( ρ ( γ )) ≥ e ¯ µ | γ | c, ∞ . Given upper domination, we have(log λ − log λ d )( ρ ( γ )) = lim n →∞ n (log σ − log σ d )( ρ ( γ n )) ≤ lim n →∞ n (log ¯ C + ¯ µ | γ n | c ) = ¯ µ | γ | c, ∞ elatively dominated representations: gaps and maps and so λ λ d ( ρ ( γ )) ≤ e ¯ µ | γ | c, ∞ . In the other direction, we remark that by [Tso20, Theorem 5.3] together withTheorem 2.9, Γ satisfies weak property U, i.e. there exist a finite subset F ⊂ Γ anda constant
L > γ ∈ Γ there exists f ∈ F with(1) | f γ | ∞ ≥ | f γ | − L. We observe that this means that given any γ ∈ Γ and ǫ >
0, there exists n > | ( f γ ) n | ≥ n | f γ | − (1 − ǫ ) Ln for all n ≥ n , or in words there is boundedcancellation between the start and end of f γ .We will now leverage this to obtain a relative version of the previous inequality,namely that for any given ǫ >
0, there exists n > | ( f γ ) n | c ≥ | f γ | c − L for all n ≥ n .To do so, we will impose some additional requirements on the finite set F ap-pearing above.To describe these requirements, and to prove our relative inequality, we will usethe framework and terminology described in § f γ to denote a geodesic path from id to f γ in the Cayley graph. Consider this f γ as a relative path ( f γ, H ) with H = H ∪ · · · ∪ H k , and write η i = f γ | H i , so each η i is a peripheral excursion. Lemma 4.3.
Given Γ a non-elementary relatively hyperbolic group, there exists afinite subset F ∈ Γ and a constant L > such that for every γ ∈ Γ there exists f ∈ F such that | f γ | ∞ ≥ | f γ | − L and the peripheral excursions of ( f γ ) n are (up to multiplicity) copies of the periph-eral excursions of f γ . We defer the proof of this statement and first complete the proof of the theoremgiven the statement.By Proposition 2.6, | f γ | c ≤ ℓ ( f γ ) − k X i =1 ℓ ( η i ) + k X i =1 ˆ ℓ ( η i ) ! . By (1), ℓ (( f γ ) n ) ≥ n | f γ | − (1 − ǫ ) Ln for all sufficiently large n . The total lengthof peripheral excursions for ( f γ ) n is at most n P ki =1 ℓ ( η i ), and, crucially, by ourassumption on the peripheral excursions of ( f γ ) n , while we might have deleted partor all of some of the peripheral excursion from f γ , none of them have merged to formlarger ones; hence the sum of the resulting ˆ ℓ is at least n P ki =1 ˆ ℓ ( η i ) − (1 − ǫ ) Ln , withequality if and only if cancellation deletes exactly (1 − ǫ ) Ln peripheral excursions,each of length 1. Now we may use Proposition 2.6 to conclude that | ( f γ ) n | c ≥ nℓ ( f γ ) − n k X i =1 ℓ ( η i ) + n k X i =1 ˆ ℓ ( η i ) − − ǫ ) Ln ! . F. Zhu
But this implies | f γ | c, ∞ = lim n →∞ n | ( f γ ) n | c ≥ ℓ ( f γ ) − k X i =1 ℓ ( η i ) + k X i =1 ˆ ℓ ( η i ) − L ! > | f γ | c − L. as desired.On the other hand it is clear that | f γ | c, ∞ ≤ | f γ | c .Now recall (for ρ a semisimple representation) there exists a finite F ′ ⊂ Γ and
C > γ ∈ Γ there exists f ′ ∈ F ′ such that for every i , | log λ i ( ρ ( γf ′ )) − log σ i ( ρ ( γ )) | ≤ C. Now, given upper eigenvalue gaps, we have σ σ d ( ρ ( γ )) ≤ C · λ λ d ( ρ ( γf )) ≤ C e ¯ µ | γf ′ | c, ∞ ≤ C e ¯ µ | γf ′ | c ≤ C ( C F ′ ) ¯ µ · e ¯ µ | γ | c where C F ′ := max f ′ ∈ F ′ e | f ′ | c and so upper domination holds. Given weak eigen-value gaps, we have σ σ ( ρ ( γ )) ≥ C − · λ λ ( ρ ( γf ′ )) ≥ C − ¯ Ce ¯ µ | γf ′ | c, ∞ ≥ C − ¯ Ce − ¯ µL e ¯ µ | fγf ′ | c ≥ C − ¯ Ce − ¯ µL ( C F C ′ F ) − ¯ µ · e ¯ µ | γ | c where C F ′ is as above and C F := max f ∈ F e | f | c , and hence (D − ) holds. (cid:3) Proof of Lemma 4.3.
We adapt the proof of [Tso20, Theorem 5.3] to show that wecan choose F to satisfy the additional requirements we have imposed here.Let f be a Floyd function f : N → R + for which the Floyd boundary ∂ f Γ of Γ isnon-trivial. By Theorem 2.9, there is a map from ∂ f Γ to the Bowditch boundary ∂ (Γ , P ) which is injective on the set of conical limit points; hence, by [Kar03],Proposition 5, we can find non-peripheral f , f such { f +1 , f − } ∩ { f +2 , f − } = ∅ .We will use sufficiently high powers of these to form our set F ; the north-southdynamics of the convergence group action of Γ on ∂ f Γ will do the rest.To specify what “sufficiently high” means it will be useful to define an auxiliaryfunction G : Z > → R > , which gives a measure of “distance to infinity” as mea-sured by the Floyd function: concretely, take G ( x ) := 10 P ∞ k = ⌊ x/ ⌋ f ( k ). Since f isa Floyd function, G ( x ) ց x → ∞ . By [Kar03, Lemma 1], we have d f ( g, h ) ≤ G ( h g, h i e ) d f ( g, g + ) ≤ G ( | g | / By the monotonicity of f and because x ∈ Z , our choice of G bounds from above the function4 xf ( x ) + 2 P ∞ k = x f ( k ) appearing in Karlsson’s proof. elatively dominated representations: gaps and maps for g, h ∈ Γ. Let ǫ = d f ( f ± , f ± ). Fix M > G ( x ) ≥ ǫ if and onlyif x ≤ M , R > G ( x ) ≤ ǫ if and only if x ≥ M , and N such thatmin {| f N ′ | , | f N ′ |} ≥ M + R ) for all N ′ ≥ N . Claim.
For every non-trivial γ ∈ H such that d f ( γ + , γ − ) ≤ ǫ , there exists i ∈ { , } such that d f ( f N ′ i γ + , γ − ) ≥ ǫ for all N ′ ≥ N . Proof of claim.
By our choice of ǫ , we can find i ∈ { , } such that d f ( γ + , f + i ) ≥ ǫ :if d f ( γ + , f ± ) < ǫ , then d f ( γ + , f ± ) ≥ d ( f ± , f ± ) − ǫ ≥ ǫ . Without loss ofgenerality suppose i = 1. There exists n such that G (cid:0) | γ n | (cid:1) < ǫ for all n ≥ n .For n ≥ n and N ′ ≥ N , by our choice of N , we have d f ( γ n , f − N ′ ) ≥ d f ( γ + , f − ) − d f (cid:16) f − , f − N ′ (cid:17) − d f ( γ + , γ n ) ≥ ǫ − G (cid:18) | f N ′ | (cid:19) − G (cid:18) | γ n | (cid:19) > ǫ. Hence, for all n ≥ n and N ′ ≥ N , G ( h γ n , f − N ′ i e ) ≥ d f ( γ n , f − N ′ ) > ǫ , and,by our choice of M , h γ n , f − N ′ i e ≤ M . Now choose a sequence ( k i ) i ∈ N such that | f k i − N | < | f k i | for all i ∈ N . For n ≥ n and N ′ ≥ N , we have, by the definitionof the Gromov product and the inequalities above,2 h f N ′ γ n , f k n i e = | f N ′ γ n | + | f k n | − | f N ′ − k n γ n | = | γ n | + | f N ′ | − h γ n , f − N ′ i e + | f k n | − | f N ′ − k n γ n |≥ | f N ′ | − M + | f k n | − ( | f N ′ − k n γ n | − | γ n | ) ≥ | f N ′ | − M + | f k n | − | f N ′ − k n |≥ | f N ′ | − M ≥ R. Then by our choice of R we have d f ( f N ′ γ + , f +1 ) ≤ lim i →∞ G ( h f N ′ γ n , f k n i i ) ≤ ǫ/ n ≥ n , N ′ ≥ N ; thus d f ( f N ′ γ + , γ − ) ≥ d f ( γ + , f +1 ) − d f ( f N ′ γ + , f +1 ) − d f ( γ + , γ − ) ≥ ǫ whence the claim. (cid:3) Now, with f , f and N as above, fix F = { f N , f N +11 , f N , f N +12 , e } . Thenthere exists g ∈ F such that d f ( gγ + , γ − ) ≥ ǫ : if d f ( γ + , γ − ) ≥ ǫ , choose g < ǫ .Otherwise, from the above argument, either g = f N or g = f N works, and then sodoes g = f N +11 or g = f N +12 (respectively.)Next fix L = 2 max g ∈ F | g | + 2 R + 1; we will show that the desired result holdswith F := F ∪ S and this L . Without loss of generality suppose | γ | > L − | γ | − | γ | ∞ ≤ L and we have our desired inequality with g = e . Otherwisechoose g ∈ F such that d f ( gγ + , γ − ) ≥ ǫ . To use this to obtain an inequalitybetween | gγ | and | gγ | ∞ , we use Lemma 2.12 with gγ in the place of γ , X theCayley graph, and x = e to obtain a sequence ( m i ) i ∈ N such that2 lim i →∞ h ( gγ ) m i , ( gγ ) − i e ≥ | gγ | − | gγ | ∞ , (3)so it suffices to obtain an upper bound on the Gromov products h ( gγ ) m i , ( gγ ) − i e . F. Zhu
To obtain this bound, we start by noting that gγ + = ( gγg − ) + , and using this,the triangle inequality, and the inequalities in (2) to observe that d f (cid:0) gγ + , ( gγ ) + (cid:1) ≤ d f (cid:0) gγ + , gγg − (cid:1) + d f (cid:0) gγg − , gγ (cid:1) + d f (cid:0) ( gγ ) + , gγ (cid:1) ≤ G (cid:18) | gγg − | (cid:19) + G (cid:0) h gγg − , gγ i e (cid:1) + G (cid:18) | gγ | (cid:19) and using liberally the monotonicity of G on the last right-hand side, we obtain thefurther upper bound d f (cid:0) gγ + , ( gγ ) + (cid:1) ≤ G (cid:18) | γ | − | g | (cid:19) which, finally, because | γ | − | g | ≥ R , is bounded above by ǫ . Arguing similarly,we have d f (cid:0) γ − , γ − g − (cid:1) ≤ d f (cid:0) γ − , γ − (cid:1) + d f (cid:0) γ − , γ − g − (cid:1) ≤ G (cid:18) | gγ | (cid:19) + G (cid:0) h γ − , γ − g − i e (cid:1) ≤ G (cid:18) | γ | − | g | (cid:19) ≤ ǫ d f (cid:0) gγ + , γ − g − (cid:1) ≥ d f (cid:0) gγ + , γ − (cid:1) − d f (cid:0) gγ + , ( gγ ) + (cid:1) − d f (cid:0) γ − , γ − g − (cid:1) ≥ ǫ − ǫ − ǫ ǫ . Thus we have n > G (cid:0) h ( gγ ) n , ( gγ ) − i e (cid:1) ≥ d f (cid:0) ( gγ ) n , ( gγ ) − (cid:1) ≥ ǫ andso h ( gγ ) n , ( gγ ) − i e ≤ M for all n ≥ n . This is the bound we feed into (3) to obtain | gγ | − | gγ | ∞ ≤ M ≤ R ≤ L , which was the inequality to be shown.Finally, we prove the statement about the peripheral excursions. To do thiswe first expand F to also include all of the non-peripheral generators in our finitegenerating set for Γ. We may also assume, without loss of generality, that gγ contains at least one peripheral excursion, otherwise there is nothing left to prove.If we have a relation αηβ with η ∈ P r { id } peripheral and α, β / ∈ P (and α notending in any letter of P and β not starting in any letter of P ), then αηα − = β − ηβ ,and by malnormality this implies α = β − , which is not possible since η = id. Sincewe are assuming gγ has peripheral excursions, we may thus assume that in ( gγ ) n there is no cancellation across more two copies of gγ , i.e. it suffices to look atcancellation between adjacent copies.We will then prove the stronger statement that the peripheral excursions of( gγ ) n are exactly n copies of that of gγ . This happens precisely when cancellationbetween adjacent copies of gγ does not reach any of the peripheral excursions.Suppose now that this is not the case, i.e. cancellation between adjacent copiesdoes reach the peripheral excursions. If g = f Ni (or g = f N +1 i , respectively), thenwe may take g = f N +1 i (respectively, g = f Ni ) instead; the desired inequalities stillhold from the arguments above, and now cancellation between adjacent copies nolonger reaches the peripheral excursions.Suppose instead g = e ; then we may assume, from the argument above, that | γ | ≤ L −
1. We will instead take g to be a non-peripheral generator s ; then, while elatively dominated representations: gaps and maps we had cancellation between adjacent copies before with g = e , we can no longerhave it with g = s . Then | sγ | ≤ | γ | + 1 ≤ L , and we are done. (cid:3) Remark 4.4.
Theorem 4.1 still holds without explicitly assuming relative hyper-bolicity, for Γ a finitely-generated group and P a collection of finitely-generatedsubgroups satisfying additional hypotheses (RH) ([Zhu19, Definition 4.1]; these hy-potheses are always satisfied when Γ is hyperbolic relative to P ) such that (Γ , P )satisfies the modified weak property U stated in Lemma 4.3.In this case we can still build X , and make the definitions and argue as above.Thus it should suffice e.g. if Γ is virtually torsion-free and admits a non-trivialFloyd boundary: the proof of Lemma 4.3 could be modified to avoid any use of theBowditch boundary, and malnormality of peripheral subgroup is one of the (RH)hypotheses (cf. [Tso20, Theorem 5.3].)We remark, though that this may be equivalent to relative hyperbolicity, seeRemark 2.10. It would in any case follow, from [Zhu19, Theorem 6.1] that if (Γ , P )admits a representation satisfying weak eigenvalue gaps (and unique limits anduniform transversality), then Γ is hyperbolic relative to P .Without relative hyperbolicity, the eigenvalue gap conditions as stated are ingeneral weaker than the analogous conditions formulated with translation lengthinstead of stable translation length: see [KP20, Example 4.8] for an explicit exam-ple. 5. Limit maps imply well-behaved peripherals
If we assume that our group Γ is hyperbolic relative to P , then the additionalconditions of unique limits and uniform transversality which appear in either ofthe definitions of relatively dominated representations so far may also be replacedby a condition stipulating the existence of suitable limit maps from the Bowditchboundary ∂ (Γ , P ). As noted above, this gives us relative analogues of some of thecharacterizations of Anosov representations due to Gu´eritaud–Guichard–Kassel–Wienhard [Gu´e+17, Theorems 1.3(1),(3) and 1.7(1),(3)]. Theorem 5.1.
Let Γ be hyperbolic relative to P . A representation ρ : Γ → SL( d, R ) is P -dominated relative to P if and only if (D − ) there exist constants ¯ C, ¯ µ > such that for all γ ∈ Γ , σ σ ( ρ ( γ )) ≥ ¯ Ce ¯ µ | γ | c , (D+) there exist constants C , µ > such that σ ( ρ ( η )) ≤ C e µ | η | c for everyperipheral element η ∈ S P , and • there exist continuous, ρ -equivariant, transverse limit maps ξ ρ : ∂ (Γ , P ) → P ( R d ) and ξ ∗ ρ : ∂ (Γ , P ) → P ( R d ∗ ) .Proof. If ρ is P -dominated relative to P , then it is assumed to satisfy (D ± ), andit admits continuous, equivariant, transverse limit maps by [Zhu19, Theorem 7.2].Conversely, if suffices to show that the unique limits and uniform transversalityconditions must hold once we have continuous, equivariant, transverse limit maps.Unique limits is necessary for the limit maps to be well-defined, since there is a singlelimit point in ∂ (Γ , P ) for each peripheral subgroup. Transversality is immediatefrom the hypotheses, and the uniform version follows from a short argument, asdone in [Zhu19, Proposition 8.5]. (cid:3) Corollary 5.2.
Given (Γ , P ) a relatively hyperbolic group, a representation ρ : Γ → SL( d, R ) is P -dominated relative to P if and only if F. Zhu • (weak eigenvalue gaps) there exist constants ¯ C, ¯ µ > such that λ λ ( ρ ( γ )) ≥ ¯ Ce ¯ µ | γ | c, ∞ for all γ ∈ Γ , • (upper eigenvalue gaps) there exist constants ¯ C, ¯ µ > such that λ λ d ( ρ ( γ )) ≤ ¯ Ce ¯ µ | γ | c, ∞ for all γ ∈ Γ , • there exist continuous, ρ -equivariant, transverse limit maps ξ ρ : ∂ (Γ , P ) → P ( R d ) and ξ ∗ ρ : ∂ (Γ , P ) → P ( R d ∗ ) .Proof. This follows from Theorem 5.1 and (the proof of) Theorem 4.1, which showsthat the eigenvalue gap conditions are equivalent to the corresponding singular valuegap conditions (D ± ). (cid:3) As an application, we can show that certain groups that play weak ping-pongon flag spaces are relatively dominated.We remark that these examples have previously been claimed in [KL18], andthat a different proof that positive representations (Example 5.6) are relativelydominated can be found in upcoming work [CZZ] of Canary–Zhang–Zimmer.
Example 5.3.
Fix biproximal elements T , . . . , T k ∈ PGL( d, R ). Write T ± i todenote the attracting lines and H ± T i to denote the repelling hyperplanes of T ± i .Assuming T + i = T + j for i = j and T ± i H ∓ T j whenever T i = T − j , and replacingthe T i with sufficiently high powers if needed, we have disjoint open neighborhoods A ± i ⊂ P ( R d ) =: X of T ± i , and B ± i ⊂ X of H ± T i such that T ± i (cid:0) X r B ± i (cid:1) ⊂ A ± i for i = 1 , . . . , k .The group Γ := h T , . . . , T k i is a non-abelian free group by a ping-pong argu-ment. There is a homeomorphism from the Gromov boundary ∂ ∞ Γ to the limitset Λ Γ ⊂ P ( R d ) and the inclusion Γ ֒ → PGL( d, R ) is P -Anosov (see e.g. [Can+17,Theorem A.2].)Now suppose we have, in addition, unipotent elements S , . . . , S ℓ ∈ PGL( d, R )which have well-defined attracting lines and attracting hyperplanes (equivalently,well-defined largest Jordan blocks). Write S + j to denote the attracting line of S j ,and suppose (again passing to sufficiently high powers if need be) S , . . . S ℓ are suchthat there exist pairwise disjoint open neighborhoods C ± , . . . , C ± ℓ whose closurescontain S +1 , . . . , S + ℓ resp., and whose closures are also disjoint from the closuresof all of the B ± i , such that S ± j ( X r C ∓ j ) ⊂ C ± j . Then, again by a ping-pongargument, the group Γ := h T , . . . , T k , S , . . . , S ℓ i is isomorphic to a non-abelianfree group F k + ℓ .Since we have finitely many generators, we can pick ǫ > • for all i = 1 , . . . , k and for any n > n < U ( T ni ) is within ǫ of T + i (resp., T − i ), • for all i = 1 , . . . , k and for any n < n > U d − ( T ni ) is within ǫ of H + T i (resp., T + i ), and • for all j = 1 , . . . , ℓ and for any n = 0, U ( S nj ) is within ǫ of S + j .By taking powers of the generators and slightly expanding the ping-pong neighbor-hoods if needed, we may assume that ǫ is sufficiently small so that the A ± i , B ± i elatively dominated representations: gaps and maps contain the 2 ǫ -neighborhoods of the T ± i and H ± T i respectively, and the C + j ∪ C − j contain the 2 ǫ -neighborhoods of the S + j . This slight strengthening of ping-pongwill be useful for establishing the transversality of our limit maps below.Let Γ ′ < Γ be the free subgroup generated by these powers.Let H j = h S j i = Stab Γ ′ S + j and let P be the set of all subgroups of Γ ′ conjugateto one of the H j . Then Γ ′ is hyperbolic relative to P and there are continuousΓ ′ -equivariant homeomorphisms ξ, ξ ∗ from the Bowditch boundary ∂ (Γ ′ , P ) to thelimit set Λ Γ ′ ⊂ P ( R d ) and the dual limit set Λ ∗ Γ ′ ⊂ P ( R d ) ∗ given by lim n γ n lim n U ( γ n ) and lim n γ n lim n U d − ( γ n ) respectively.We claim that ξ and ξ ∗ are transverse: given two distinct points x = lim γ n and y = lim η n in ∂ (Γ , P ), we have ξ ( x ) / ∈ ξ ∗ ( y )—the latter considered as a hyperplanein P ( R d )—using ping-pong and the following Lemma 5.4 ([Gu´e+17, Lemma 5.8]; [BPS19, Lemma A.5]) . If A, B ∈ GL( d, R ) are such that σ p ( A ) > σ p +1 ( A ) and σ p ( AB ) > σ p +1 ( AB ) , then d ( B · U p ( A ) , U p ( BA )) ≤ σ σ d ( B ) · σ p +1 σ p ( A ) . To establish the claim: write γ n = g · · · g n and η n = h . . . h n . Pick n minimalsuch that U ( γ n ) and U ( η n ) are in different ping-pong sets. The lemma aboveimplies that for any given ǫ >
0, there exists some n so that for all n ≥ n , U ( γ n ) = U ( g · · · g n ) is ǫ -close to γ n · U ( g n +1 · · · g n ), and U d − ( η n ) is ǫ -close to η n · U d − ( h n +1 · · · h n ). By our ping-pong set up, for sufficiently small ǫ these areuniformly close to U ( γ n ) and U d − ( η n ) respectively, and in particular they areseparated from each other.Moreover, the inclusion ι : Γ ֒ → PGL( d, R ) satisfies (D ± ).(D+) is immediate from Γ ′ being finitely generated, the existence of a polynomial p of degree d − σ σ d ( u ) ≤ p ( | u | )) for every unipotent element u ∈ Γ ′ , andthe sub-multiplicativity of the first singular value σ .To obtain (D-), one can use the following Lemma 5.5 ([BPS19, Lemma A.7]) . If A, B ∈ GL( d, R ) are such that σ p ( A ) >σ p +1 ( A ) and σ p ( AB ) > σ p +1 ( AB ) , then σ p ( AB ) ≥ (sin α ) · σ p ( A ) σ p ( B ) σ p +1 ( AB ) ≤ (sin α ) − σ p +1 ( A ) σ p +1 ( B ) where α := ∠ ( U p ( B ) , S d − p ( A )) . To use this here, we show that if ( γ n = g · · · g n ) n ∈ N ⊂ Γ is a sequence convergingto ∈ ∂ (Γ , P ), where each g i is a power of a generator and g i and g j are not powersof a a common generator whenever | i − j | = 1, then ∠ ( U p ( g · · · g i − ) , S d − p ( g i )) isuniformly bounded from below for p = 1 , d − α > s and divergentsequences ( k n ) of integers and ( w n ) of words in Γ not starting in s ± (assumefurther—passing to a subsequence if needed—that these words converge to somepoint in ∂ (Γ , P )) such that ∠ ( U ( ρ ( w n )) , S d − ( ρ ( s k n ))) ≤ − n . Then, in the limit, we obtain ∠ (cid:16) lim n →∞ U ( ρ ( w n )) , lim n →∞ S d − ( ρ ( s k n )) (cid:17) = 0 F. Zhu but this contradicts transversality, since by our hypothesis that none of the words w n starts with s , we must have lim n →∞ w n = lim n →∞ s k n .Thus we do have a uniform lower bound α ≤ α as desired, and then Lemma5.5, together with the explicit estimate σ σ ( S nj ) ≥ q ( | u | ))for some non-constant polynomial q , tells us that log σ p σ p +1 grows at least linearlyin | γ n | c , which gives us (D − ). We remark that this part of the argument does notrequire the strengthening to the ping-pong described above.We then conclude, by Theorem 5.1, that ι : Γ ֒ → PGL( d, R ) is P -dominatedrelative to P . Example 5.6.
Let Σ be a surface with boundary and write Γ = π Σ, Let ρ : Γ → PSL( d, R ) be a positive representation in the sense of Fock–Goncharov [FG06], andlet P be the collection of cyclic subgroups of Γ corresponding to holonomies ofboundary components with unipotent image under ρ .By [FG06, Theorem 1.9], any element of Γ not in a conjugate of one of thesubgroups in P is positive hyperbolic; as a consequence, Γ is hyperbolic relative to P . Moreover, suppose we put a hyperbolic metric on Σ, such that the boundarycomponents with unipotent holonomy are represented by punctures, and the oneswith non-unipotent holonomy by geodesic boundary components. This gives us amap ι ∞ : ∂ (Γ , P ) ֒ → ∂ ∞ H ∼ = R P identifying ∂ (Γ , P ) with a subset of R P .By [BT18, Theorem 4.9] (via collapsing all of the boundary components of Σto punctures), ρ admits a Schottky presentation in F + ( R d ), the space of orientedflags on R d , and hence, by [BT18, Theorem 2.10], we can find a left-continuousequivariant increasing boundary map ξ : R P → F + ( R d ).We can compose this boundary map with the natural projections from F + ( R d ) to P ( R d ) and P ( R d ) ∗ to obtain (left-continuous) equivariant limit maps from R P tothose respective Grassmannians. Moreover, by considering the opposite orientationon R P , we get right-continuous equivariant limit maps, and by construction thesewill agree on on ι ∞ ( ∂ (Γ , P )) ⊂ R P , thus giving us continuous equivariant limitmaps from ∂ (Γ , P ) to the respective Grassmannians.These maps are transverse from the increasing condition: if x and y are disjointpoints in R P , let z be such that x, z, y are in cyclic order; then ξ ( x ) , ξ ( z ) , ξ ( y ) areoriented flags in cyclic order, meaning they form a 3-hyperconvex triple of orientedflags, in the sense of [BT18] (see Proposition 3.7 and Definition 3.1 there): inparticular, the flags are a fortiori pairwise transverse.As in the previous example, we can obtain (D ± ) from suitable linear algebra andping-pong. Thus, again by Theorem 5.1, ρ is P -dominated relative to P .Indeed, such ρ are P k -dominated relative to P for all 1 ≤ k ≤ d −
1, as definedin [Zhu19], analogous to how purely hyperbolic Schottky groups are Borel–Anosov[BT18, Theorem 1.3].
References [Bow12] B. H. Bowditch. “Relatively hyperbolic groups”. In:
International Jour-nal of Algebra and Computation doi : . elatively dominated representations: gaps and maps [BPS19] Jairo Bochi, Rafael Potrie, and Andr´es Sambarino. “Anosov representa-tions and dominated splittings”. In: J. Eur. Math. Soc. (JEMS) doi : .[BT18] Jean-Philippe Burelle and Nicolaus Treib. “Schottky groups and maxi-mal representations”. In: Geom. Dedicata
195 (2018), pp. 215–239. doi : .[Can+17] Richard D. Canary, Michelle Lee, Matthew Stover, and Andr´es Sam-barino. “Amalgam Anosov representations”. In: Geometry and Topology doi : .[CDP90] M. Coornaert, T. Delzant, and A. Papadopoulos. G´eom´etrie et th´eoriedes groupes . Vol. 1441. Lecture Notes in Mathematics. Les groupes hy-perboliques de Gromov. [Gromov hyperbolic groups], With an Englishsummary. Springer-Verlag, Berlin, 1990, pp. x+165. isbn : 3-540-52977-2.[CZZ] Richard Canary, Tengren Zhang, and Andrew Zimmer.
Anosov repre-sentations of geometrically finite Fuchsian groups . In preparation.[DGO17] F. Dahmani, V. Guirardel, and D. Osin. “Hyperbolically embedded sub-groups and rotating families in groups acting on hyperbolic spaces”. In:
Mem. Amer. Math. Soc. doi : .[FG06] Vladimir Fock and Alexander Goncharov. “Moduli spaces of local sys-tems and higher Teichm¨uller theory”. In: Publications Math´ematiquesde l’Institut des Hautes ´Etudes Scientifiques doi : .[Flo80] William J. Floyd. “Group completions and limit sets of Kleinian groups”.In: Invent. Math. doi : .[Ger12] Victor Gerasimov. “Floyd maps for relatively hyperbolic groups”. In: Geom. Funct. Anal. doi : .[GM08] D. Groves and J. F. Manning. “Dehn filling in relatively hyperbolicgroups”. In: Israel Journal of Mathematics doi : .[GP13] Victor Gerasimov and Leonid Potyagailo. “Quasi-isometric maps andFloyd boundaries of relatively hyperbolic groups”. In: J. Eur. Math.Soc. (JEMS) doi : .[Gro13] Bradley W. Groff. “Quasi-isometries, boundaries and JSJ-decompositionsof relatively hyperbolic groups”. In: J. Topol. Anal. doi : .[Gro87] M. Gromov. “Hyperbolic groups”. In: Essays in group theory . Vol. 8.Math. Sci. Res. Inst. Publ. Springer, New York, 1987, pp. 75–263. doi : .[Gu´e+17] Fran¸cois Gu´eritaud, Olivier Guichard, Fanny Kassel, and Anna Wien-hard. “Anosov representations and proper actions”. In: Geom. Topol. doi : .[Kar03] Anders Karlsson. “Free subgroups of groups with nontrivial Floyd bound-ary”. In: Comm. Algebra doi : .[KL18] M. Kapovich and B. Leeb. Relativizing characterizations of Anosov sub-groups, I . June 2018. arXiv: . F. Zhu [KLP17] Michael Kapovich, Bernhard Leeb, and Joan Porti. “Anosov subgroups:dynamical and geometric characterizations”. In:
Eur. J. Math. doi : .[KLP18a] Michael Kapovich, Bernhard Leeb, and Joan Porti. “A Morse lemma forquasigeodesics in symmetric spaces and Euclidean buildings”. In: Geom.Topol. doi : .[KLP18b] Michael Kapovich, Bernhard Leeb, and Joan Porti. “Dynamics on flagmanifolds: domains of proper discontinuity and cocompactness”. In: Geom. Topol. doi : .[KP20] Fanny Kassel and Rafael Potrie. Eigenvalue gaps for hyperbolic groupsand semigroups . July 2020. arXiv: .[Lev20] Ivan Levcovitz. “Thick groups have trivial Floyd boundary”. In:
Proc.Amer. Math. Soc. doi : .[Rie21] Max Riestenberg. A quantified local-to-global principle for Morse quasi-geodesics . Jan. 2021. arXiv: .[Tso20] Konstantinos Tsouvalas.
Anosov representations, strongly convex co-compact groups and weak eigenvalue gaps . Aug. 2020. arXiv: .[Zhu19] Feng Zhu.
Relatively dominated representations . Dec. 2019. arXiv:1912.13152 [math.GR]