Let G be a discrete finitely generated group acting and properly discontinuously by isometries on a space X in one of the following situations:
(1) G acts properly discontinuously by isometries on a CAT(-1) space X , (2) G is a lattice in a higher Lie group G, acting on its symmetric space X .
(3) G is the mapping class group of a surface acting on its Teichmüller space.
Let G ⊂ ∂ X denote the limit set-the collection of accumulation points of an(y) orbit on the boundary ∂ X .
The aim of this paper is to establish a connection between three perspectives on the action of G on G pertaining to three different themes as mentioned below:
(1) maxima of stationary symmetric α-stable (SαS) random fields indexed by G ( Probability Theory), (2) extreme values of cocycles given by Radon-Nikodym derivatives of Patterson-Sullivan measures induced by the quasi-invariant action of G on its limit set G ⊂ ∂ X ( Ergodic Theory), (3) extrinsic geometry of the orbit of G on X in terms of whether the Bowen-Margulis measure is finite or not ( Non-positively curved and Hyperbolic Geometry).
The relation between ( 1) and ( 2) has been studied in probability in the context of abelian G and free G. The relation between ( 2) and (3) on the other hand has been studied thoroughly in the context of pairs (X , G) as above. However the connection between (1) and ( 3) is unexplored territory for pairs (X , G) as above. We achieve this connection in the present paper via the mediation of ergodic theoretic techniques (2), which play a key role in the proofs of our main results. One of the main tools we use from ergodic theory is mixing of the geodesic flow with respect to the Bowen-Margulis measure. The basic test case where G is a free group and X its Cayley graph with respect to a standard generating set had been dealt with in [40]; however this example is somewhat orthogonal to the main thrust of the present paper and examples explored therein, as geodesic flow is not mixing in the case of the free group. To address this largely excluded case of the free group, we devote a final subsection to normal subgroups of free (or more generally hyperbolic) groups, where the Bowen-Margulis measure is used and we recover the corresponding theorem from [40].
The connection between the probabilistic and the ergodic theoretic perspectives ((1) and (2) in the above list) is, in the general form that constitutes the background of this paper, due essentially to Rosiński [34][35][36] (see also the encyclopedic monograph [41] and the recent survey [37]). The study of stationary SαS random fields (i.e., stochastic processes indexed by G such that each finite linear combination follows an SαS distribution) is important in probability theory because such fields appear as scaling limits of regularly varying random fields having various dependence structures. These random fields come naturally equipped with a Rosiński representation, thus connecting with measurable dynamical systems in a canonical manner. The naturality of SαS random fields in the context of dynamical/ergodic-theoretic applications is in fact a consequence of the exact correspondence, furnished by the Rosiński representation, between such stochastic processes and quasi-invariant (or nonsingular) group actions, and hence dynamical cocycles. We outline the connection in Sect. 2 and summarize the discussion as follows (for details, see Theorem 2.4).
Given a standard measure space (S, μ) equipped with a quasi-invariant (i.e., measure-class preserving) group action {φ g } g∈G , a ±1-valued cocycle {c g } g∈G (that is, c gh (s) = c h (s) + c g (φ h (s))
for {φ g } and a function f ∈ L α (S, μ), there exists a stationary SαS random field {Y g } indexed by G admitting an integral representation (known as the Rosiński representation):
where the above integral is with respect to an SαS random measure M on S with control measure μ. We recall that a random measure M is called an SαS random measure with control measure μ if for each set A with μ(A) < ∞, the random variable M(A) follows an SαS distribution with scale parameter (μ(A)) 1/α ; see, for example, [41].
Conversely, given a stationary SαS random field {Y g } indexed by G, there exist a standard measure space (S, μ) equipped with a quasi-invariant group action {φ g } g∈G , a ±1-valued cocycle {c g } g∈G and a function f ∈ L α (S, μ) such that Y g admits a Rosiński representation given as above.
When μ is a probability measure (often the case in this paper), we shall use to denote the space S (as our probability measures will be typically supported on limit sets ). With this change of notation, the basic probabilistic question we address in this paper is: Question 1.1 Find sufficient conditions on a non-singular conservative action of G on a probability measure space ( , S, μ) to ensure that the growth of partial maxima of the associated stationary SαS random field indexed by G is like the i.i.d. case.
When G = Z d and X is a Cayley graph of G with respect to a standard generating set, this can never happen [38,39]. There is only one recent example giving a positive answer to Question 1.1: G = F d is free, X is a Cayley graph of G with respect to a standard generating set, and is the Cantor-set boundary of F d equipped with the Patterson-Sullivan measure [40]. In this paper we prove that there is a large class of examples, geometric in origin, giving a positive answer to Question 1.1 (see Theorem 6.1):
(1) Non-elementary Gromov-hyperbolic groups G acting on a Cayley graph X = G (with respect to a finite generating set) and on the boundary = ∂G, equipped with the Patterson-Sullivan measure class. This directly generalizes the main theorem of [40]. (2) Groups G acting on proper CAT(-1) spaces X with limit set ⊂ ∂ X equipped with the Patterson-Sullivan measure class, and satisfying the following:
• The length spectrum of X /G is non-arithmetic,
• the associated Bowen-Margulis measure on the unit tangent bundle U M of M = X /G is finite.
(3) A lattice G in a higher rank lie group, acting on the symmetric space X and its Furstenberg boundary = ∂ X equipped with the Patterson-Sullivan measure class. (4) The mapping class group G acting geometrically on Teichmüller space X and measurably on the Thurston boundary = ∂ X equipped with the Thurston measure.
For groups G acting on proper CAT(-1) spaces X with limit set ⊂ ∂ X (as in Item (2) above), finiteness of the Bowen-Margulis measure μ B M of U M in fact provides a new phase transition boundary for the behavior of the growth of partial maxima. The growth of partial maxima is like the i.i.d. case if and only if the Bowen-Margulis measure μ B M of U M is finite (Theorem 6.1 Item(2) and Theorem 6.3). An important technical tool that we use in the proofs of the main Theorems 6.1 and Theorem 6.3 is mixing of the geodesic flow (in cases (2), (3), (4) above). Mixing of the geodesic flow in turn is used to count the number of orbit points inside an n-ball.
Mixing, in this strong form, fails for Gromov hyperbolic groups equipped with the word metric [4]. However, for infinite normal subgroups of infinite index in such groups, we establish a slightly weaker counting technique for the number of orbit points. This allows us to obtain Theorem 6.7: the behavior of partial maxima for a normal subgroup H of a hyperbolic group G is i.i.d. -like if and only if H is of finite index in G. In the setup of hyperbolic groups, the latter provides the analog of Theorem 6.3 -the μ B M (U M) = ∞ case for CAT(-1) spaces.
A key aim in this paper is to bring into focus the geometry underpinning Question 1.1. We replace the default word metric of earlier works on the subject [38][39][40] by a general proper geodesic metric space (X , d). Apart from Roblin's fundamental dichotomy on the behavior of the Poincaré series [33], the tools we bring in to answer Question 1.1 are also from the more geometric aspects of ergodic theory: mixing of the geodesic flow and equidistribution of spheres. We introduce an invariant called extremal cocycle growth incorporating both the geometry of the action of G on (X , d) as well as the quasi-invariant action of G on ( , S, μ) whose asymptotic qualitative behavior determines the answer to Question 1.1. This invariant records the appropriately normalized maximal distortion of the measure μ at a point ξ ∈ with respect to actions of group elements g which move a fixed point o ∈ X a bounded amount.
Throughout (X , d) will be a proper geodesic metric space. Definition 1.2 [14] Let Ĉ(X ) = C(X )/ ∼ denote the compact set of 1-Lipschitz functions with the topology of convergence on compact subsets, where
Given a (parameterized) geodesic ray γ ⊂ (X , d), the Busemann function based at x for γ is given by
If X is CAT(0), i(X ) = X equals the usual visual compactification X and ∂ h X is the visual boundary. If X = T eich(S) is the Teichmüller space, i(X ) gives the Gardiner-Masur compactification (see [23] and references therein).
If (X , d) is Gromov-hyperbolic and ξ is an endpoint of a geodesic γ , the equivalence class of β γ 's with γ (∞) = ξ will be denoted as β ξ . Also, if (X , d) is Gromovhyperbolic, quotienting ∂ h X further by bounded functions we obtain the Gromov boundary ∂ g X [6,Section 2.5]. The pre-image of ξ (under this further projection) are the elements of the equivalence class β ξ . To get a well-defined Busemann function in this case, instead of an equivalence class, we shall define
(1.2)
In all three cases (CAT(0), Teichmüller, or Gromov-hyperbolic) we shall choose a base-point o and normalize Busemann functions such that β ξ (o) = 0. If in addition X is a Cayley graph of a hyperbolic group, o will be the identity.
Now suppose G acts properly discontinuously by isometries on X .
For X Gromov-hyperbolic, let M(∂ g X ) denote the collection of positive finite Borel measures on
for all g ∈ G.
If Eq. 1.3 holds for some C > 0 and all x, y ∈ X , ξ ∈ ∂ X we shall simply write
omitting the specific value of C.
The setup for the rest of the paper is as follows. (G, X , ) will denote one of the following:
(1) X is a proper CAT(-1) space, G a non-elementary discrete group acting properly discontinuously by isometries on X (recall that G is non-elementary means that its limit set is infinite), and the limit set. (2) X is a proper Gromov-hyperbolic space, G a non-elementary discrete group acting properly discontinuously by isometries on X , and the limit set. (3) X is the Teichmüller space T eich(S) of a closed surface S, G = MCG(S) acting on T eich(S), and the Thurston Boundary PMF (S). (4) X is a symmetric space for a Lie group G of higher rank, G is a lattice in G, and the Furstenberg boundary of X .
We shall use the convention that ∂ X stands for the horofunction boundary ∂ h X in cases 1, 3, 4. and the Gromov boundary ∂ g X in Case 2. In all the cases μ will denote a conformal or quasiconformal density on ⊂ ∂ X (see Sect. 3.1 for existence).
Let (X , G, ) be as in Setup 1.4. Let x → μ x be a G-invariant conformal or quasiconformal density of dimension v. Define V n = ex p(vn).
(
In this paper we shall be interested in the dichotomy given by zero and non-zero extremal cocycle growth. It will suffice therefore to estimate the asymptotics of the RHS of Eq. (1.8).
A common inspiration for many of the settings we study in this paper is the action of the group SL(2, Z) on the hyperbolic plane X = H 2 . The limit set of SL(2, Z) is R ∪ ∞, and the action on the boundary (and on H 2 ) is given by fractional linear maps,
where g = a b c d . The Lebesgue measure class is preserved, and if we choose standard Lebesgue measure as our reference measure μ, the Radon-Nikodym derivative
If we take, for example our basepoint o in X = H 2 as i, and our radius n = log 3, there are 5 elements of SL(2, Z) (we are eliding the issue of elements that stabilize i, here) which have d(g.o, 0) ≤ n, namely the identity, and the matrices
If we take ξ = 2 as our reference point on the boundary, we have that the Radon-Nikodym derivatives 1 (2c + d) 2 which takes on the values 1 and 1/3 at the matrices above, so A log 3 (2) = 1, in this formulation. The group SL(2, Z) can also be viewed as acting on its Cayley graph, which gives a different interpretation which is also generalized in our work.
We point out here that the free group on 2 generators F 2 provides us with examples to which both Theorems 6.1 and 6.3 apply. The group G = F 2 thus furnishes three kinds of examples of non-singular conservative actions.
(1) F 2 acting on its own boundary equipped with the Patterson-Sullivan measure.
Here, extremal cocycle growth is non-vanishing and hence the growth of partial maxima of the associated SαS random field is like the i.i.d. case. (2) F 2 may be identified with an index 6 subgroup of P SL(2, Z) (the second congruence subgroup). Hence, as in Sect. The above examples illustrate that the growth of partial maxima of the associated SαS random field does not depend on the group G alone but rather on the geometry of the space X on which it acts, and via this action, on the specific nature of the associated probability measure space ( , S, μ) on which G admits a non-singular action.
In Sect. 2, we give a brief review of group indexed SαS-random fields {X g } g∈G and deduce a basic criterion (Theorem 2.6) in terms of non-vanishing or vanishing of ECG (Definition 1.5) that determines whether the partial maxima of {X g } g∈G exhibits i.i.d. -like behavior or not. This reduces the purely probabilistic question 1.1 to the following question lying at the interface of geometry, dynamics and probability: Question 1.6 Find sufficient conditions on triples (G, X , ) such that ECG is nonvanishing.
In Sect. 3, we recall various theorems from the literature that show that Patterson-Sullivan measures in the context of Setup 1.4 give quasiconformal densities. We also recall work of Furman [13] and Bader-Furman [4] on Bowen-Margulis measures. In Sect. 4 we recall results on mixing of the geodesic flow and establish consequences on convergence of spherical averages. In the special case of the mapping class group acting on Teichmüller space, the corresponding result (Theorem 4.4) appears here for the first time. Section 5 is the technical core of the paper and relates spherical averages to ECG. In Sect. 6 we prove the main Theorems of the paper. Theorem 6.1 establishes non-vanishing of ECG in the four cases mentioned at the beginning of the Introduction and Theorem 6.3 establishes vanishing of ECG for CAT(-1) examples with infinite Bowen-Margulis measure. Normal subgroups of hyperbolic (e.g. free) groups are treated in Sect. 6.3.
We shall use (S, S, μ) to denote a σ -finite general Borel measure space and ( , S, μ) to denote a probability measure space. Definition 2.1 A (real-valued) random variable Y is said to follow a symmetric αstable (SαS) distribution with tail parameter α ∈ (0, 2] and scale parameter σ > 0 if it has characteristic function of the form
The value of the tail parameter equal to two corresponds to the Gaussian case. Here we shall largely focus on α ∈ (0, 2), i.e. the non-Gaussian case (see [41] for a detailed treatment of α-stable (0 < α < 2) distributions).
Let G be a finitely generated infinite group with identity element e. A random field (that is, a collection of random variables)
Integral Representations Any such random field has an integral representation of the type
where M is an SαS random measure on some σ -finite standard Borel space (S, S, μ), and f g ∈ L α (S, μ) for all g ∈ G; see Theorem 13.1.2 of [41]. This simply means that each linear combination k i=1 c i Y g i follows an SαS distribution with scale parameter k i=1 c i f g i α . We shall always assume, without loss of generality, that
We shall simply write stationary to mean left-stationary throughout this paper. [34][35][36] Given a standard measure space (S, μ) equipped with a quasi-invariant group action {φ g } g∈G , a ±1-valued cocycle {c g } g∈G for {φ g }, and an f ∈ L α (S, μ), there exists a stationary SαS random field indexed by G admitting an integral representation (known as the Rosiński representation):
Conversely, given a stationary SαS random field {Y g } indexed by G, there exist a standard measure space (S, μ) equipped with a quasi-invariant group action {φ g } g∈G , a ±1-valued cocycle {c g } g∈G and an f ∈ L α (S, μ) such that Y g admits a Rosiński representation given by (2.2).
The stationary random field indexed by G corresponding to the standard measure space (S, μ), the quasi-invariant group action {φ g } g∈G , the ±1-valued cocycle {c g } g∈G and an f ∈ L α (S, μ) is denoted as
In the special case that f ≡ 1 (here we must have μ(S) < ∞), we simplify the notation to
If (S, μ) is a probability measure space, we shall replace S by .
In this work, we are interested in the rate of growth of the partial maxima sequence
as n increases to ∞. It was shown in [38,39] that when G = Z d , the rate of growth of M n is like the i.i.d. case if and only if the action of Z d on (S, μ) in Theorem 2.4 above is not conservative. In general, the rate of growth of M n is controlled by that of the deterministic sequence
See, for example, Section 3 of [39]. The following proposition relates this sequence with the extremal value of the cocycle defined in (1.6).
be a stationary SαS random field indexed by G where μ is a quasiconformal measure supported on ⊂ ∂ X . Then b n = A n 1/α , where A n is as in (1.6).
Proof Incorporating the form of Rosiński representation in b n , we get
as in Proposition 2.5. Let
Then the following dichotomy holds:
(1) If the action of G on ( , μ) has non-vanishing extremal cocycle growth, then given any subsequence of {M n }, there exists a further subsequence
where d → denotes convergence in distribution. Further, Z α is a Frechét type extreme value random variable and κ is a positive constant that may depend on the choice of the subsequence {M n k }. If further lim n→∞ C n exists (and hence is positive), then M n /V 1/α n converges weakly to the limit in (2.4).
(2) If the action of G on ( , μ) has vanishing extremal cocycle growth, then The key ingredient of the proof of Theorem 2.6 is the following series representation (see, for instance, Equation (4.12) of [39]): For all fixed n ≥ 1,
where ( 1) " d =" denotes equality of distribution, (2) b n 's are given by Eq. ( 2.3), ( 3)
(4) ε j 's are i.i.d. Bernoulli random variables taking ±1 values with equal probability, (5) {U (n) j : j ≥ 1} is an i.i.d. sequence of -valued random variables with common law given by
for all j ≥ 1, j = E 1 + E 2 + • • • E j with E j 's being i.i.d. exponential random variables with unit mean.
Note that the right hand side of (2.6) converges almost surely, and the equality of distribution can be verified for each linear combination of the two sides with the help of Theorem 1.4.2 of [41].
We split into two cases: Case 1: ECG is non-vanishing For simplicity, let us assume that lim n→∞ C n exists (not just the limit inferior) and hence is positive. With this assumption,
converges to a positive constant and as in the proof of Equation (4.9) in [39] (see for instance the heuristics below), it follows that
for some c > 0. This completes a (sketch of a) proof of the last statement in Theorem 2.6.
In the general case, lim inf n→∞ C n exists and is positive. Hence given any subsequence of {C n }, there is a further subsequence {C n k } that converges to a positive limit. Therefore, applying the argument used above on this subsequence, we obtain (2.4). Now, by Theorem 3.1 of [40] Case 1 follows.
Case 2: ECG is vanishing The proof of (2.5) relies on a comparison argument given in [39] (see the proof of Equation (4.3) therein). As in Example 5.4 of [39] and Example 6.1 of [40], we construct an auxiliary stationary SαS random field {Y g } g∈G with Rosiński representation given by
on a standard probability space ( , μ ) such that
for some a > 0 and ∈ (0, 1/α). Without loss of generality, we may assume that and are disjoint sets. We consider the stationary SαS random field
which has a canonical Rosiński representation on ∪ with the action being {φ g } restricted to ( , μ) and {φ g } restricted to ( , μ ). Therefore, the {b Z n } sequence corresponding to {Z g } satisfies
because of vanishing of ECG and (2.8). Using the inequality b Z n ≥ a|B n | , the series representation (2.6) and the arguments given in the proof of (4.3) in [39], it follows that M n /b Z n is stochastically bounded (also known as tight), i.e., given any η ∈ (0, 1) there exists
Heuristics and idea behind (2.7): Instead of rewriting in detail the proof of Equation (4.9) in [39], we provide the heuristics behind it. The main tool for verifying (2.7) is, as expected, the series representation (2.6) mentioned above. The heuristics behind this are based on the one large jump principle, which can be described as follows. It can be shown that
See Pages 26-28 of [41]. According to the discussion on Page 26 of this reference, the first term of (2.6) is the dominating term that gives the precise asymptotics of its tail while the rest of the terms provide the "necessary corrections" for the whole sum to have an SαS distribution.
In light of the above one large jump heuristics, we get that for all λ > 0,
This computation yields (2.7). The key step (namely, the "≈" above) can be made precise with the help of (2.6) and the language of Poisson random measures; see Pages 1454-1455 of [39] for details.
is stochastically bounded (also known as tight) and is "bounded away from zero" as long as ECG is non-vanishing.
Let (X , d) be a proper geodesic metric space with base-point o and equipped with a properly discontinuous isometric action of a group G. In all the cases of interest in this paper, there exists v > 0, called the critical exponent such that for all s > v, the Poincaré series converges and for all s < v, the Poincaré series diverges.
We refer the reader to [11,30,42] for the construction of Patterson-Sullivan measures when X is Gromov-hyperbolic. The limit set G (⊂ ∂ X ) of the group G acting on X is the collection of accumulation points in ∂G of a G-orbit G.o for some (any) o ∈ X . The group G acts by homeomorphisms on G given by φ g (x) = g -1 • x. We shall represent this action as g -→ φ g . This is consistent with the action in Eq. 1.5:
Theorem 3.2 [11] Let (X , d) be a proper Gromov-hyperbolic metric space equipped with a properly discontinuous (not necessarily convex cocompact) isometric action of a group G. Then there exists a quasiconformal density of dimension v (equal to the critical exponent) supported on the limit set = G . Also, v = lim sup n
The quasiconformal density constructed by Coornaert in Theorem 3.2 is called the Patterson-Sullivan density. When P s (x, y) diverges, the Patterson-Sullivan measure based at o is obtained as a weak limit of the measures g∈B n e -sd(x,g.y) Dirac g.y normalized by P s (x, y) (see [6,9] for details). When P G (s) converges, an extra weighting function is introduced in front of the exponential factors to force the modified P s (x, y) to diverge [30]. Note that V n (Eq. 1.4) can be identified with the volume growth of balls of radius n in the weak hull C H( ) of in X , where C H( ) consists of the union of geodesics with end-point in .
In this subsection, X will be a symmetric space of noncompact type and G a lattice. The visual or geometric boundary will be denoted as ∂ X , while the Furstenberg boundary will be denoted as ∂ F X . The critical exponent of the Poincaré series is denoted by v as before. The Furstenberg boundary ∂ F X can be naturally identified with the orbit of the centroid of a Weyl chamber in ∂ X . Thus, ∂ F X ⊂ ∂ X . As before, the action of g on ∂ F X will be denoted by g → φ g .
Albuquerque shows (see Definition 1.3):
Theorem 3.3 [2] For (X , G) as above, there exists a unique conformal density given by the Patterson-Sullivan measure class {μ ξ } supported on ∂ F X ⊂ ∂ X.
In this subsection, X will denote the Teichmüller space T eich(S) of a surface and G = MCG(S) its mapping class group. The Thurston boundary, or equivalently, the space PMF (S) of projectivized measured foliations, will be denoted as ∂ X . Let ξ ∈ ∂ X be a measured foliation. Let E xt ξ (x) denote the extremal length at x ∈ X of a measured foliation ξ . The Thurston measure on the space of measured foliations MF(S) is denoted as μ. For ξ ∈ MF(S), [ξ ] will denote its image in PMF (S).
For U ⊂ PMF (S), the authors of [1] define a measure μ x with base-point x ∈ X as follows:
Further, [1,p. 1064]
.
For [ξ ] ∈ PMF (S), a Busemann-like cocycle β ξ : T eich(S) × T eich(S) → R for the Teichmüller metric is defined as follows:
This makes the family {μ x } x∈X of probability measures on PMF (S) into a family of G-invariant conformal densities of dimension v = dim(X ) for the cocycle β:
Theorem 3.4 [1] Let (X , G), μ x , v be as above. For all x, y ∈ X and
Further, for g ∈ G, U ⊂ PMF (S)
We shall refer to any μ x above as a Thurston conformal density.
As observed at the end of Section 2.3 of [1], it follows from [1,Theorem 2.9] that the family of measures μ x give a Patterson-Sullivan density on ∂ X .
We shall recall the construction of Bowen-Margulis measures by Furman [13] and Bader-Furman [4] for X Gromov-hyperbolic and G acting properly discontinuously by isometries on it. This is slightly more general than what we need in most of the applications. We will apply it in particular to CAT(-1) spaces (see [33] for an excellent treatment in the latter context). Let
G × R given by s (x, y, t) = (x, y, t + s). G × R, μ B M S × L) given by (x, y, t) → g • (x, y, t) satisfying the following:
(1) G preserves the infinite measure μ B M S × L.
(2) The G-action commutes with the R -action.
(3) The G-action commutes with the flip: (x, y, t) → (y, x, -t).
We call (U M, μ B M ) the measurable unit tangent bundle corresponding to the action of G on X ; and μ B M the Bowen-Margulis measure on the measurable unit tangent bundle U M.
The two variable growth function is given as follows:
We refer the reader to [33] for an excellent introduction to Patterson-Sullivan and Bowen-Margulis measures in the context of CAT(-1) spaces. We shall say that a group action of G on X has non-arithmetic length spectrum if there does not exist c > 0 such that all translation lengths are integral multiples of c. Roblin [33] proved the following dichotomy for group actions on C AT (-1) spaces.
(1) There exists a function c G : X × X → R + such that V G (x, y, n) c G (x, y)e vn if the Bowen-Margulis measure of the measurable unit tangent bundle is finite:
The proof of Theorem 3.8 above depends crucially on mixing of the geodesic flow in this context [33,Chapter 3]. In this strong form, it fails for hyperbolic groups equipped with the word metric (see the discussion after [4,Corollary 1.7], where the authors prove a weaker version of mixing).
Let ( , m) be a finite measure space. Let G be a a locally compact topological group acting on (X , m) preserving m. The G-action is said to be mixing if, for any pair of measurable subsets A, B ⊂ , and any sequence
For this subsection X is a proper CAT(-1) space, and M = X /G. We can construct the geometric tangent bundle to M as follows:
where G acts diagonally. In this context, Roblin [33] constructs the Bowen-Margulis measure μ B M on U g M converting it to a space measure-isomorphic to the measurable unit tangent bundle (U M, μ B M ) (described just after Proposition 3.7).
When μ B M (U M) < ∞, Roblin [33, Chapter 3] proves that the Bowen-Margulis measure μ B M is mixing under the geodesic flow on U M unless the length spectrum is arithmetic (see also [18,32]). Conjecturally, arithmetic length spectrum is equivalent to the condition that there exists c > 0 such that X is isometric to a tree with all edge lengths in cN (this has been proven under the additional assumption that the limit set of G is full, i.e. G = ∂ X in [32]). Let P : U M → M be the natural projection, so that P -1 ( p) = S p may be thought of as the 'unit tangent sphere' at p ∈ M. S p can be naturally identified with the boundary ∂ X equipped with the Patterson-Sullivan measure μ supported on the limit set G (see for instance the discussion on skinning measures in [31, Section 3]). We denote this measure by μ p and think of it as the Patterson-Sullivan measure on G based at p. Broise-Alamichel, Parkkonen and Paulin [3,31] (see also [12]) prove that when X is CAT(-1) and the geodesic flow is mixing, then μ p equidistributes to the Bowen-Margulis measure (see also [28,29] where the considerably more general notion of skinning measures was introduced). Let A ⊂ U M denote any measurable subset and let A p,t := {x ∈ S p |g t (x) ∈ A}. We summarize these results below: Theorem 4.1 [3,12,31,32] Suppose X is C AT (-1) such that M = X /G has nonarithmetic length spectrum. For U M as above, suppose μ B M (U M) < ∞. Then the sphere S( p, t) of radius t about a point p ∈ X /G becomes equidistributed as t → ∞ in the following sense. For any measurable subset A ⊂ U M, and p ∈ M,
as t → ∞.
Using results of Kleinbock-Margulis [15,16], we will prove the following general statement:
Thus, we have Theorem 4.2. In fact [15] gives a precise estimate on the rate of convergence, linking it to the exponential rate of mixing for the flow g t . In fact, this rate of convergence can be bounded below for any g t = exp(tz) where z is in the norm 1 subset of the positive Weyl chamber A + , and so by doing an extra integration over this set, we can get equidistribution of the whole sphere in the space M. See [16, §6] for more details on describing the geodesic flow on symmetric spaces using the orbits of one-parameter subgroups, following ideas of Mautner.
For the purposes of this subsection, let X = T eich(S) be the Teichmüller space of a surface S, G = MCG(S) be its mapping class group, and M = X /G the moduli space. Let U X (resp. U M) denote the bundle of unit-norm holomorphic quadratic differentials on X (resp. M). Let π : U M → M denote the natural projection. Let g t denote the Teichmüller geodesic flow on U M. Masur-Smillie [24] building on earlier work of Masur [19] and Veech [45] showed that U M carries a unique measure μ (up to scale) in the Lebesgue measure class such that μ(U M) < ∞ and g t is mixing. Let η = π * μ denote push-forward of μ to M. For any x ∈ M, denote the unit-norm holomorphic quadratic differentials at x by S(x). Identify S(x) with the Thurston boundary PML(S) of T eich(S) and equip it with the Thurston measure ν x based at x. Let η t,x = π * g * t ν x denote the measure ν x pushed forward to the sphere of radius t {π(g t (x, v))} ⊂ M, i.e.,
To prove this theorem, fix (x, v) ∈ U M. The group SL(2, R) acts on U X, and its action commutes with the mapping class group, so it acts on U M. The action of the group
is precisely the geodesic flow g t . The circles {a t r θ (x, v) : 0 ≤ θ ≤ 2π }, where andlet dκ x,v (r θ (x, v)) = dκ(θ),and let κ x,v,t = a * t κ(x, v). To prove Theorem 4.4, we will use the following ergodic theorem of Nevo's [27, Theorem 1.1], which in our case implies:
To use this theorem for our result, we note that there is a measure ω on S(x) = S(x)/K so that we can write
We also note that the function f (x, v) = d(π(x, v), x 0 ) is K -invariant, and by the following lemma of Masur [20], in L 2 (U X, μ). Let (x, v) denote the length of the shortest saddle connection of (x, v) (recall that a saddle connection is a geodesic in the flat metric on S determined by the quadratic differential (x, v) joining two zeroes, with no zeroes in its interior).
There is a constant C such that for any x 0 ∈ M and (x, v) ∈ U M, we have
By Masur-Smillie [24],
which, combined with the lemma, yields that f ∈ L 2 (U X, μ). To finish the proof Theorem 4.4, we note that if there was a positive η-measure set of x ∈ M so that the set of
f dμ had positive ω-measure, we would have a set of positive μ-measure in U X where (4.1) fails, a contradiction.
In this section, we shall establish a connection between extremal cocycle growth as in Definition 1.5 and the asymptotics of spherical averages. This will, in particular, allow us to apply the equidistribution theorems of the previous section.
Let G, X , μ be as in Setup 1.
We also have the natural projections π r : r (o) → ∂ X .
Definition 5.1 Let X be CAT(0) or T eich(S). A sequence of probability measures {μ r } on r (o) is said to be a sequence of averaging measures with respect to a conformal density μ supported on ⊂ ∂ X if
(1) π r ,t * (μ r ) = μ t , for r > t, (2) there exists C ≥ 1 such that if μ ∞ (a measure on ∂ X ) is any weak limit of π r * (μ r ) up to subsequences, then μ ∞ is supported on and 1/C ≤ dμ ∞ dμ (ξ ) ≤ C, ∀ξ ∈ . If C = 1, then {μ r } is said to be strongly averaging.
When X is a uniformly proper δ-hyperbolic graph with all edges of length one, π rt is not well-defined, but only coarsely so. Thus, for x ∈ r (o), we define π rt (Dirac x ) to be the uniform probability distribution on the set
Note that for r > t, the support of π rt (Dirac x ) has diameter at most δ. Definition 5.2 Let X be a uniformly proper δ-hyperbolic graph with all edges of length one. A sequence of probability measures {μ r } on r (o) is said to be a sequence of averaging measures with respect to a quasiconformal density μ supported on ⊂ ∂ X if there exists C ≥ 1 such that
Note that in Definitions 5.1 and 5.2, the projections π sr for fixed r and s > r can be extended to a projection π r : → r (o) such that Let v denote the dimension of the conformal or quasiconformal density μ on . For f 0 : M → R + given by f 0 (w) = ex p(-vd M (y 0 , w)), f ex = ex p(-v d X (x, G.o)) will denote the lift of f 0 to X . The spherical averages {E r ( f ex )} for (X , G, f ex ), given by
will be called extremal spherical averages with respect to the base-point o.
Note that the domain of f ex is X (and hence it can be integrated over r (o)).
Recall that V r = e vr (Eq. 1.4) and B r = {g ∈ G|g.o ∈ B(o, r )}. We write B r .o = {g.o |g ∈ B r } and define:
The domain of f exr is X ∪ ∂ X and hence it can be integrated over both s (o) and G . Define
f exr (x) dμ r (x), (
We shall write
Proof The second inequality is clear and we need only to prove the first. We use the fact that the measure μ in all the examples in 5.3 are Patterson-Sullivan densities.
In particular, it follows from the construction of Patterson-Sullivan measures that for every k ∈ N, there exists C 0 ≥ 1 such that if ν k is any limit of uniform probability distributions on the k-shells (r ,k) as r → ∞, then 1/C 0 ≤ dν k dμ (ξ ) ≤ C for all ξ ∈ . Hence the uniform probability distributions on the 'inner' k-shell (r -k,k) and the 'outer' k-shell (r ,k) about r (o) are close to each other: more precisely any two limits of uniform distributions on (r -k,k) and (r ,k) are absolutely continuous with respect to each other with pointwise Radon-Nikodym derivative lying in
Now, we use the fact that {μ r } is a sequence of averaging measures. We argue by contradiction. Suppose that for some fixed m, no K ≥ 1 exists as in the conclusion of the Lemma. We pass to the limit as r → ∞. Extracting subsequential limits if necessary, there exists a limit μ -of the inner shell measures, a limit μ + of the outer shell measures and a measurable subset U ⊂ such that μ + (U ) = 0 while μ -(U ) > 0. This contradicts the absolute continuity in the last sentence of the previous paragraph, proving the Lemma.
Corollary 5.7 Let {μ r } be averaging measures on r (o) with respect to μ as in 5.3. For any > 0 there exists K , R 0 ≥ 1 such that for r ≥ R 0 ,
Proof Choose k ∈ N such that e -kv ≤ < e -(k-1)v and let m = k. For x ∈ r (o), f ex (x) ≥ implies d(x, B r +k .o) ≤ m. The Corollary now follows from Lemma 5.6.
We are now in a position to state a sufficient condition guaranteeing non-vanishing ECG.
Proposition 5.8 Let (G, X , ) be as in cases 1,3, or 4 of setup 1.4 with M = X /G and o ∈ X a base-point. Let P : X → M be the quotient map, P(o) = y 0 . Let {μ r } be averaging measures on r (o) with respect to μ as in 5.3. Suppose that there exists c, R 0 > 0 and 1 ≥ α > 0 such that for all r ≥ R 0 ,
Then the action of G on ( , μ) has non-vanishing extremal cocycle growth.
Proof Let = e -cv . It follows by hypothesis, that for all r ≥ R 0 ,
Hence by Corollary 5.7, there exists K ≥ 1 such that for all r ≥ R 0
Since {μ r } is a family of averaging measures, there exists K 1 , R 1 ≥ 1 such that for all r ≥ R 1 , μ( (r , ) ≥ α/K 1 . Hence, from Eq. 1.8, there exists
Thus, lim inf r →∞ C r > 0; equivalently, the action of G on ( , μ) has non-vanishing extremal cocycle growth.
When X is Gromov-hyperbolic, i.e. Cases 1, 2 of Setup 1.4 we can say more. Proposition 5.9 Let (G, X , ) and μ be as in Cases 1, 2 of Setup 1.4. Let {μ r } be a family of averaging measures as in 5.3. Then there exists R 0 such that for r ≥ R 0 , C r f exr (q)dμ(q) r (o)
f exr (x)dμ r (x).
(5.5)
To prove Proposition 5.9 we shall need the following basic Lemma from hyperbolic geometry (see, for instance, [22,Lemma 3.3] or [21,Lemma 3.3] for a proof): Lemma 5.10 Given δ, C ≥ 0 there exists C such that the following holds: Let X be a δ-hyperbolic metric space and K ⊂ X be C-quasiconvex. For any p ∈ X \ K , let π K ( p) denote a nearest point projection of p onto K and let [ p, π K ( p)] be the geodesic segment joining p, π K ( p). For k ∈ K , let [π K ( p), k] be the geodesic segment joining π
For q ∈ and q ∈ [o, q), π r (q ) being uniformly close to π r (q), we have the following consequence of Lemma 5.10 using the fact that the balls B(o, r ) are δ-quasiconvex. For q ∈ ∂ X and a ∈ X , the geodesic ray from a to q is denoted as [a, q). Corollary 5.11 Given δ > 0 there exists C > 0 such that the following holds for any r > 0: If X is a δ-hyperbolic metric space, W r = QC(
We restate the last statement of Corollary 5.11 in the form that we shall use, unwinding the definition of the Busemann function β q (o, k) based at q: Corollary 5.12 Given δ > 0 there exists C > 0 such that the following holds for any r > 0: if X is a δ-hyperbolic metric space, W r = QC(
Proof of Proposition 5.9: By Corollary 5.12 we have,
Since V r = ex p(vr ), we have
Hence,
This proves the first asymptotic equality of Proposition 5.9. A standard argument using the Sullivan shadow lemma (see for instance [11,Proposition 6.1] or [10,Proposition 3.11]) shows that the projection π r : → r (o) and the shadow map from r (o) to may be used as approximate inverses of each other for large r . Hence, integrals over r (o), equipped with the averaging measure μ r , converge, up to uniform multiplicative constants, to the integral over ( , μ). Thus, there exists R 0 > 0 such that for r ≥ R 0 ,
f exr dμ r (x), completing the proof of Proposition 5.9.
In this section, we shall prove the main theorems of the paper.
Theorem 6. 1 The following triples (G, X , ) have non-vanishing extremal cocycle growth:
(1) (X , d) is a proper Gromov-hyperbolic metric space equipped with a properly discontinuous convex cocompact isometric action of a group G. The limit set of G is equipped with a Patterson-Sullivan measure μ. (2) (X , d) is a proper complete CAT(-1) space equipped with a properly discontinuous isometric action of a group G such that M = X /G has non-arithmetic length spectrum. The limit set of G is equipped with a Patterson-Sullivan measure μ. Further, assume that for M = X /G, the Bowen-Margulis measure
and μ is the Thurston conformal density based at a generic base-point o ∈ X (i.e. o belongs to a full measure subset of M = X /G). (4) (X , d) is a symmetric space of non-compact type equipped with a properly discontinuous isometric action of a lattice G. The limit set is the Furstenberg boundary ∂ F X embedded canonically in ∂ X as the K-orbit of the barycenter of a Weyl chamber at infinity [2]. The limit set of G is equipped with the Patterson-Sullivan measure μ.
Hence, in all the above cases, the associated group indexed stationary random fields (via the Rosiński representation) {Y g := Y g ( , μ, {φ g }, {c g }), g ∈ G, } is i.i.d. -like (see Definition 2.7) with respect to the behavior of partial maxima.
Proof We give a case-by-case argument: Item 1: This will follow immediately from Proposition 5.9 if we can prove that f exr (x) is uniformly bounded below point-wise on r (o) (independent of r ). Since f exr = ex p(-vd X (x, B r .o)), the point-wise lower bound on f exr will follow from a pointwise upper bound on d X (x, B r .o) for x ∈ r (o). But this is an immediate consequence of the fact that G acts on X cocompactly. The last statement of Theorem 6.1 now follows from Theorem 2.6.
An alternate argument for Item 2 above can be given by directly invoking Roblin's Theorem 3.8 for the asymptotics of V G (x, y, n) when μ B M (U M) < ∞.
However the proof here generalizes directly to Items 3, 4.
The purpose of this subsection is to prove: Proof Let f exr be as in Proposition 5.9. Since m r (C) → 0 as r → ∞, it follows that for all > 0, there exists N ∈ N such that for all r ≥ N , m r (C) < , and
Hence by Proposition 5.9,
Since can be made arbitrarily small and C arbitrarily large, lim r →∞ C r = 0.
When X is the Cayley graph of a free group with respect to a standard set of generators, Item 2 of Theorem 6.1 does not apply as the geodesic flow is not mixing in this case (mixing fails more generally a hyperbolic group equipped with the word metric [4]).
We deal in this section with subgroups H of hyperbolic groups G, especially when H is normal. For the purposes of this subsection, X = will be a Cayley graph of G with respect to a finite set of generators. It follows immediately from Theorem 6.1 that if H is a finite index subgroup of G, then the action of H on (∂G, μ P S ) has non-vanishing extremal cocycle growth. Assume henceforth that H is an infinite index subgroup of G. H is said to be co-amenable in G if the left action of G on the (right) coset space /H is amenable. We shall use: Theorem 6.5 [7] For G, H , X (= ) as above, let v G and v H denote the exponential growth rates of G and H acting on X . Then H is co-amenable in G if and only if
Let V H ( When the Poincaré series for the H -action on converges at v H = v G , then also, V H (1, 1, n) = o(ex p(v G n)). The proof of Theorem 6.3 again shows that ECG is vanishing in this case.
We finally come to: Theorem 6.7 Let G be a hyperbolic group and H an infinite normal subgroup of infinite index. Let X = denote a Cayley graph of G with respect to a finite generating set. Let μ P S denote the Patterson-Sullivan measure of H on the limit set H = ∂G. Then the ECG for the triple (X , H , H ) vanishes.
Proof It suffices, by Proposition 6.6, to assume that v H = v G = v and that the Poincaré series of H diverges at v. Further, as in the proof of Theorem 6.3, it is enough to show that V H (1, 1, m) = o(ex p (vm)).
We now invoke a Theorem due to Matsuzaki, Yabuki and Jaerisch [25,Theorem 4.2], [26,Theorem 1.2] that ensures that the Patterson-Sullivan measure μ P S of H is, up to uniformly bounded multiplicative constants, invariant under the action of G. We normalize so that μ P S is constructed with base-point 1 ∈ H . Thus, for all g ∈ G, g * μ P S μ P S g -1 ,
where the suffix g -1 indicates the shifted base-point. Hence, for all q ∈ H = ∂G, g * μ P S (q) + (g -1 ) * μ P S (q) μ P S (q), where indicates uniform multiplicative constants independent of g. Since G/H is infinite, we can choose distinct g 1 , g -1 1 • • • , g n , g -1 n such that for all q ∈ H = ∂G, n 1 [g * i μ P S (q) + (g -1 i ) * μ P S (q)] nμ P S (q).
As usual, let B(1, m) denote the m-ball in X = . It follows that for any distinct g 1 , g -1 1 • • • , g n , g -1 n , there exists N such that for m ≥ N ,
Since
it follows that for m ≥ N ,
Since n can be made arbitrarily large, V H (1, 1, m) = o(ex p (vm)) as required.
(1) Replacing Albuquerque's results [2] in Sect. 3.1.2 by a Theorem of Link [17,Theorem A] gives immediately an analog of Theorem 6.1 Item (4) for lattices in products of negatively curved manifolds.
(2) The proof of Item 1 of Theorem 6.1 goes through without modification when G acts cocompactly on (X , d) when the latter is only quasi-ruled in the sense of [5,Section 1.7] instead of being a geodesic metric space. Thus, let X = (G, S) where G is hyperbolic and S is a finite generating set. Let μ be a finitely supported symmetric measure on G whose support generates G. Let ν be the hitting measure on ∂G. Let d be the Green metric on X [5] and let = (∂G, ν). Then proof of Item 1 of Theorem 6.1 goes through and shows that the action of G on has non-vanishing ECG. Hence by Theorem 2.6, the behavior of partial maxima is i.i.d. -like. Note that in this case, the Busemann function is computed with respect to the Green metric rather than the word metric [5].
(3) An exact analog of Ricks' theorem [32] on mixing and convergence of spherical averages 4.1 is absent at this point for general Gromov-hyperbolic spaces. This is the only obstruction in obtaining an exact analog of Theorem 6.1 Item (2) for general Gromov-hyperbolic spaces.
(4) Dependence on α: For a stationary SαS random field Y g = Y g (S, μ, φ g , c g , f ), ECG (Definition 1.5) identifies the qualitative behavior of partial maxima when μ is a probability measure and f is a constant function. Note that this qualitative behavior (of being i.i.d. -like) is independent of α. Thus, to determine the dependence of partial maxima on α, we really need to investigate the general case of non-constant f .
(5) It might be worthwhile to extract axiomatically the essential features from all the examples of non-vanishing ECG in Theorem 6.1 to provide a general sufficient condition. (6) It was kindly pointed out to us by the referee that the main result of the paper [8] by Coulon-Dougall-Schapira-Tapie generalizes Theorem 6.5 as follows. In Theorem 6.5, X is assumed to be a Cayley graph of G. However, [8] allows Theorem 6.5 to go through when G acts cocompactly on X , or more generally when X /G has finite Bowen-Margulis measure. This allows all the results for subgroups of hyperbolic groups to go through in this more general context. We hope to take up some of the unexplored issues in the above list in subsequent work.