Recurrence Properties of Random Walks on Finite Volume Homogeneous Manifolds

Alex Eskin^{∗} and Gregory Margulis^{†}

March 16, 2002

1 Introduction

Let G be a semisimple Lie group, and Γ a nonuniform irreducible lattice in G. Let π denote the natural projection from G to G/Γ. We recall some results from [Dan1-Dan4], [Mar]. Let U = {u_{t}} be a unipotent one-parameter subgroup of G.

Theorem 1.1. For every point x ∈ G/Γ and every ϵ > 0 there exists a compact set K ⊂ G/Γ such that for all T > 0

|{t ∈ [0, T] : u_{t}x ∈ K}| > (1 − ϵ)T  (1)

More generally, for every compact set C ∈ G/Γ and every ϵ > 0 there exists a compact set K such that (1) holds for all x ∈ C, and all T > 0.

A parabolic subgroup P of G is called Γ-rational if the unipotent radical of P intersects Γ in a lattice. Note that if Γ is arithmetic then a parabolic subgroup of G is Γ-rational if and only if it is defined over ℚ.

Theorem 1.2. There exists a compact set K with the following property: Let g ∈ G be any element such that gUg^{−1} is not contained in a Γ-rational parabolic subgroup of G. Then Uπ(g) intersects K.

∗Research partially supported by NSF grant DMS-9704845, the Sloan Foundation and the Packard Foundation

†Research partially supported by NSF grant DMS-9800607

Remark 1.3. Essentially the assertion of Theorem 1.2 is that every orbit of U returns to a fixed compact set K. An example that the algebraic condition on gUg^{−1} in Theorem 1.2 is needed can be given as follows: Let G be SL(n, ℝ) and Γ = SL(n, ℤ). Suppose gUg^{−1} is contained in a ℚ-parabolic P. Let V be the subspace stabilized by P. Since V is defined over ℚ, V intersects ℤ^{n} in a lattice L. Note that the volume of the torus V/V ∩ L is an invariant of the orbit; if it is sufficiently small, the orbit will not intersect the fixed compact set K.

We emphasize that the key point of Theorem 1.1 and Theorem 1.2 is that Theorem 1.1 holds for every point x, and Theorem 1.2 holds for every element g satisfying an explicit algebraic condition. The fact that the theorems hold almost everywhere is an easy consequence of ergodicity and the Birkhoff ergodic theorem. The proofs rely heavily on the polynomial nature of the unipotent flow, and are based mostly on the techniques of [Mar].

The aim of this paper is to show that similar results hold under some conditions when considering random walks on G/Γ. Let μ be a measure on G satisfying the condition

∫_{G} ∥g∥^{δ} dμ(g) < ∞  (2)

for sufficiently small δ > 0, where ∥ · ∥ denotes some norm in some finite dimensional representation of G. To state the results we formulate some properties analogous to the conclusions of Theorem 1.2 and Theorem 1.1 which may be possessed by the random walk defined by μ.

Definition 1.4. Let μ be a probability measure on G satisfying (2). Let μ^{(m)} denote the convolution of μ with itself m times, and for x ∈ G let δ_{x} denote the probability measure supported on the point x. The following may hold:

(R1) For every compact set C ⊂ G/Γ and every ϵ > 0 there exists a compact set K ⊃ C such that for every x ∈ C and every m > 0,

(μ^{(m)} ∗ δ_{x})(K) > (1 − ϵ)

(R2) For every ϵ > 0 there exists a compact set K such that for every x ∈ G/Γ, there exists M = M(x) > 0 such that for m > M(x),

(μ^{(m)} ∗ δ_{x})(K) > (1 − ϵ).

The constants M(x) can be chosen so that for every compact subset C of G/Γ, sup_{x∈C} M(x) < ∞.

(S) For every ϵ > 0 there exists a compact set K such that for every g ∈ G either g(suppμ)g^{−1} is contained in a Γ-rational parabolic P or there exists M > 0 such that for m > M, (μ^{(m)} ∗ δ_{π(g)})(K) > (1 − ϵ).

The property (R1) is a version of the conclusion of Theorem 1.1 which makes sense in the current context. The property (S) has the same relation to the conclusion of Theorem 1.2. We note that (R2) implies (R1) and (S).

2 Recurrence properties of random walks

To simplify the terminology, we assume that G is a connected algebraic group. All the claims can be easily reduced to this case.

Notation: Let H_{μ} denote the noncompact part of the Zariski closure of the subgroup generated by the support of μ. (By the noncompact part of an algebraic group H we mean the subgroup of H generated by unipotent elements, together with the split part of H).

Our result is the following:

Theorem 2.1. Suppose H_{μ} is semisimple, and for all g ∈ G, gH_{μ}g^{−1} is not contained in any proper Γ-rational parabolic subgroup of G. Then μ has properties (R1) and (R2). In particular, if H_{μ} = G, then μ has properties (R1) and (R2).

Recall that a measure ν on a G-space X is μ-stationary if an only if μ ∗ ν = ν. We note the following general lemma (proved in §5):

Lemma 2.2. Let μ be any measure on G satisfying (R2). Then any μ-stationary locally finite measure on G/Γ is finite.

As a corollary of Theorem 2.1 and Lemma 2.2 we have the following result conjectured by N. Shah in [Sh]:

Theorem 2.3. Let G be a semisimple Lie group, and Γ a nonuniform irreducible lattice in G. Let Λ ⊂ G be a countable Zariski dense subgroup (or more generally a countable subgroup with semisimple Zariski closure which is not contained in any conjugate of a proper Γ-rational parabolic subgroup of G). Let x ∈ G/Γ be a point such that the orbit Λx ⊂ G/Γ is discrete. Then Λx is finite.

Proof of Theorem 2.3. Since Λ is countable, we can find a measure μ such that suppμ = Λ, and (2) holds. Let σ denote counting measure on the discrete set Λx (i.e. σ(C) is the cardinality of C ∩ Λx). Then σ is invariant, hence stationary. Then, in view of Theorem 2.1 and Lemma 2.2, σ is finite. Hence Λx is finite.

As another corollary, we obtain another proof, in a special case, of the following well known theorem of Borel and Harish-Chandra:

Theorem 2.4. Let H be a semisimple algebraic group defined over ℚ. Then, H(ℤ) is a lattice in H(ℝ) (i.e. H(ℝ)/H(ℤ) has finite H(ℝ)-invariant measure).

We give a proof of this theorem under the assumption that there exists a faithful representation of H which is defined over ℚ and is irreducible over ℝ.

Proof of Theorem 2.4 (under the assumption). Let H = H(ℝ), Λ = H(ℤ). We are assuming that there is a faithful representation ρ : H → SL(N, ℝ) which is irreducible over ℝ and defined over ℚ. Since ρ is defined over ℚ, we have, after possibly replacing Λ by a subgroup of finite index, ρ(Λ) ⊂ SL(N, ℤ). Hence, we have ρ(H)/ρ(Λ) ⊂ SL(N, ℝ)/SL(N, ℤ). Let σ denote the ρ(H)-invariant measure on ρ(H)/ρ(Λ). Then σ is locally finite. Let μ be any compactly supported absolutely continuous measure on H. Then, by Theorem 2.1, and since ρ is irreducible over ℝ, ρ(μ) has property (R2). Also since σ is ρ(H)-invariant, it is also ρ(H)-stationary. Thus by Lemma 2.2, σ is finite. Hence Λ is a lattice in H.

We also make the following conjecture:

Conjecture 2.5. Suppose H_{μ} is semisimple (or more generally generated by unipotent elements). Then μ has properties (R1) and (S).

Theorem 2.1 can be generalized somewhat: see Proposition 2.6 below.

The set Δ. We now define a certain finite collection Δ of maximal parabolic subgroups of G. If G has real rank 1, then since Γ is non-uniform there exists a Γ-rational minimal parabolic subgroup P_{0} of G, and we define Δ = {P_{0}}. If the real rank of G is at least 2 then Γ is arithmetic, hence there exists a ℚ-structure on G such that Γ is commensurable with the set G(ℤ) of integer points (with respect to this ℚ-structure). We fix a maximal ℚ-split torus A_{0} for G, and we let Δ = {P_{1}, …, P_{r}} where the P_{k} are the standard parabolic subgroups with respect to A_{0}. For every P_{i} ∈ Δ, there exists a representation ρ_{i} : G → GL(V_{i}) and vectors w_{i} ∈ V_{i} such that the stabilizer of ℝw_{i} is P_{i}.

Condition A. We say that μ satisfies Condition A if it satisfies (2) and for all sufficiently small δ > 0 there exist c < 1 and n > 0 such that for all i, 1 ≤ i ≤ r and all v ∈ Gw_{i},

∫_{G} \frac{1}{∥ρ_{i}(g)v∥^{δ}} dμ^{(n)}(g) ≤ \frac{c}{∥v∥^{δ}}  (3)

Theorem 2.1 follows immediately from the following two propositions:

Proposition 2.6. Suppose μ satisfies Condition A. Then μ has properties (R1) and (R2).

Proposition 2.7. Suppose H_{μ} is semisimple and for any g ∈ G, gH_{μ}g^{−1} is not contained in any Γ-rational parabolic subgroup of G. Then μ satisfies Condition A.

Proposition 2.6 will be proved in §3 and Proposition 2.7 will be proved in §4. Lemma 2.2 will be proved in §5.

3 Systems of inequalities

In this section we prove Proposition 2.6. The proof is based on the following:

Lemma 3.1. Suppose that there exists a positive continuous function u : G/Γ → ℝ with the following properties:

(i) u(x) → ∞ as x → ∞ in G/Γ.

(ii) There exist constants c_{1} < 1 and b > 0 and n > 0 such that for any x ∈ G/Γ,

∫_{G} u(gx) dμ^{(n)}(g) ≤ c_{1}u(x) + b.  (4)

Then μ has property (R2).

Proof. We note that it is enough to prove (R2) for m in an arithmetic progression. After iterating (4) and summing the geometric series, we obtain for any multiple m of n,

∫_{G} u(gx) dμ^{(m)}(g) ≤ c_{1}^{m/n}u(x) + b_{1},  (5)

where b_{1} = b/(1 − c_{1}). Note that for any R > 0 the set {y ∈ G/Γ : u(y) < R} is compact. We choose K = {y ∈ G/Γ : u(y) < 2b_{1}/ϵ}; then, for m large enough so that c_{1}^{m/n}u(x) < b_{1},

\frac{(2b_{1})}{ϵ}(μ^{(m)} ∗ δ_{x})(K^{c}) ≤ ∫_{G} u(hx) dμ^{(m)}(h) ≤ c_{1}^{m/n}u(x) + b_{1} ≤ 2b_{1},

hence (μ^{(m)} ∗ δ_{x})(K^{c}) < ϵ as required.

3.1 Construction of the function u in the SL(d, ℝ)/SL(d, ℤ) case.

In this case the construction is more transparent, and follows closely that of [EMM]. The representation ρ_{i} of condition A is the representation of G = SL(d, ℝ) on the i’th exterior power of ℝ^{d}, and we can take w_{i} = e_{1} ∧ ⋯ ∧ e_{i}, where {e_{1}, …, e_{d}} is the standard basis for ℝ^{n}. We now recall some notation and results from [EMM]:

Let Δ be a lattice in ℝ^{d}. We say that a subspace L of ℝ^{d} is Δ-rational if L ∩ Δ is a lattice in L. For any Δ-rational subspace L, we denote by d_{Δ}(L) or simply by d(L) the volume of L/(L ∩ Δ). Let us note that d(L) is equal to the norm of u_{1} ∧ ⋯ ∧ u_{ℓ} in the exterior power ⋀^{ℓ}(ℝ^{d}) where ℓ = dim L, (u_{1}, ⋯, u_{ℓ}) is a basis over ℤ of L ∩ Δ, and the norm on ⋀(ℝ^{d}) is induced from the Euclidean norm on ℝ^{d}. If L = {0} we write d(L) = 1. A lattice is Δ unimodular if d_{Δ}(ℝ^{d}) = 1. The space of unimodular lattices is canonically identified with SL(d, ℝ)/SL(d, ℤ).

We introduce the following notation:

α_{i}(Δ) = sup \{\frac{1}{d(L)}| L is a Δ-rational subspace of dimension i \}, 0 ≤ i ≤ d,
α(Δ) = max_{0≤i≤d} α_{i}(Δ).  (6)

Lemma 3.2 ([EMM], Lemma 5.6). For any two Δ-rational subspaces L and M

d(L)d(M) ≥ d(L ∩ M)d(L + M).  (7)

In view of (2) we may write μ^{(n)} = μ_{1} + μ_{2}, where μ_{1} has compact support and for any 1 ≤ i ≤ d − 1,

∫_{G} ∥ρ_{i}(g)∥^{δ} dμ_{2}(g) ≤ \frac{1 − c}{3}  (8)

Lemma 3.3. Suppose μ satisfies condition A, and let n, δ and c be as in (3). Let μ_{1} < μ^{(n)} be any compactly supported measure. Then there exists a constant ω > 1 such that for any lattice Λ in ℝ^{d}, and any 1 ≤ i ≤ d − 1,

∫_{G} α_{i}(gΛ)^{δ} dμ_{1}(g) < c α_{i}(Λ)^{δ} + ω^{2δ} max_{0<j≤min\{d−i,i\}} (\sqrt{α_{i+j}(Λ)α_{i−j}(Λ)} )^{δ}. (9)

Proof. Let Λ be a lattice in ℝ^{d}, and let M be a Λ-rational subspace of ℝ^{d}. Then, for any g ∈ G, gM is also a gΛ-rational subspace. By (3) applied to v_{1} ∧ ⋯ ∧ v_{l} where (v_{1}, ⋯, v_{ℓ}) is a basis over ℤ of M ∩ Λ, we have

∫_{G} \frac{1}{d_{gΛ}(gM)^{δ}} dμ_{1}(g) ≤ c\frac{1}{d(M)^{δ}} .  (10)

There exists a Λ-rational subspace L_{i} of dimension i such that

\frac{1}{d_{Λ}(L_{i})} = α_{i}(Λ).  (11)

The inequality (10) implies

∫_{G} \frac{dμ_{1}(g)}{d_{gΛ}(gL_{i})^{δ}} < c\frac{1}{d_{Λ}(L_{i})^{δ}} .  (12)

Let ω = max_{0<j<d^{g∈suppμ_{1}}} ∥ρ_{j}(g)^{±1}∥. We have that for any Λ-rational subspace L, and any

g ∈ suppμ_{1},

ω^{−1} ≤ \frac{d_{gΛ}(gL)}{d_{Λ}(L)} ≤ ω.  (13)

Let us denote the set of Λ-rational subspaces L of dimension i with d_{Λ}(L) < ω^{2}d_{Λ}(L_{i}) by Ψ_{i}. We get from (13) that for a Λ-rational i-dimensional subspace L ∉ Ψ_{i}

d_{gΛ}(gL) > d_{gΛ}(gL_{i}),  g ∈ supp μ_{1}.  (14)

It follows from (12), (14) and the definition of α_{i} that

∫_{G} α_{i}(gΛ)^{δ} dμ(g) < cα_{i}(Λ)^{δ}  if Ψ_{i} = {L_{i}}.  (15)

Assume now that Ψ_{i} ≠ {L_{i}}. Let M ∈ Ψ_{i}, M ≠ L_{i}. Then dim(M +L_{i}) = i+j, j > 0. Now using (11), (13) and Lemma 3.2 we get that for any g ∈ suppμ_{1},

α_{i}(gΛ) < ωα_{i}(Λ) = \frac{ω}{d_{Λ}(L_{i})} < \frac{ω^{2}}{\sqrt{d_{Λ}(L_{i})d_{Λ}(M)}}

≤ \frac{ω^{2}}{\sqrt{d_{Λ}(L_{i} ∩ M)d_{Λ}(L_{i} + M)}}

≤ ω^{2}\sqrt{α_{i+j}(Λ)α_{i−j}(Λ)}.

(16)

Hence if Ψ_{i} ≠ {L_{i}}

∫_{G} α_{i}(gΛ)^{δ} dμ_{1}(g) ≤ ω^{2δ} max_{0<j≤min\{d−i,i\}} (\sqrt{α_{i+j}(Λ)α_{i−j}(Λ)} )^{δ}.  (17)

Combining (15) and (17) we get that for any lattice Λ ⊂ ℝ^{n}, (9) holds.

Lemma 3.4. For sufficietly small ϵ > 0 and δ > 0 the function

u(Λ) = ∑_{i=0}^{d} ϵ^{i(d−i)}α_{i}(Λ)^{δ},  (18)

satisfies the conditions of Lemma 3.1.

Proof. Note that u(Λ) satisfies (i) of Lemma 3.1. Let δ_{0} > 0 be some choice of δ satisfying (2) and Condition A, and let δ = δ_{0}/d. Since for any 1 ≤ j ≤ d − 1 and any g ∈ G, ∥ρ_{j}(g)∥ ≤ ∥g∥^{d}, we have for some constant C,

u(gΛ) ≤ C∥g∥^{dδ}u(Λ).

Hence, in view of (2), we may decompose μ^{(n)} = μ_{1} + μ_{2} such that μ_{1} is compactly supported and

∫_{G} u(gΛ) dμ_{2}(g) ≤ \frac{1 − c}{3}u(Λ),  (19)

where c < 1 and n > 0 are as in (3). Let A_{1} denote the averaging operator on G/Γ given by

(A_{1}f)(Λ) = ∫_{G} f(gΛ) dμ_{1}(g).  (20)

Let q(i) = i(d − i). Then by direct computations 2q(i) − q(i + j) − q(i − j) = 2j^{2}. Therefore we get from Lemma 3.3 that for any i, 0 < i < d, and any positive ϵ < 1

A_{1}(ϵ^{q(i)}α_{i}^{δ}) < c ϵ^{q(i)}α_{i}^{δ} + ω^{2} max_{0<j≤min\{d−i,i\}} ϵ^{q(i)− \frac{q(i+j)+q(i−j)}{2}} \sqrt{ϵ^{q(i+j)}α_{i+j}^{δ}ϵ^{q(i−j)}α_{i−j}^{δ}}

≤ c ϵ^{q(i)}α_{i}^{δ} + ϵ ω^{2} max_{0<j≤min\{d−i,i\}} \sqrt{ϵ^{q(i+j)}α_{i+j}^{δ}ϵ^{q(i−j)}α_{i−j}^{δ}}.

(21)

Since ϵ^{q(i)}α_{i}^{δ} < u, α_{0} = 1 and α_{d} = 1/d(Δ) = 1, the inequalities (21) imply the following inequality:

(A_{1}u)(Λ) < 1 + 1 + c u(Λ) + dϵω^{2}u(Λ).

Taking ϵ = \frac{1−c}{3dω^{2}} we see that

(A_{1}u)(Λ) = ∫_{G} u(gx) dμ_{1}(Λ) < \frac{1 + 2c}{3}u(Λ) + 2  (22)

Now, in view of (22) and (19), (ii) of Lemma 3.1 also holds. This completes the proof of Lemma 3.4, and hence the proof of Proposition 2.6 in the SL(d, ℝ)/SL(d, ℤ) case.

3.2 Construction of the function u in the general case.

Let P_{0} denote the minimal Γ-rational parabolic subgroup of G. Then we have the Langlands decomposition P_{0} = M_{0}A_{0}N_{0}, where M_{0} is semisimple, N_{0} is the unipotent radical of P_{0}, and A_{0} is as in the definition of Δ. From the general theory, Γ ∩ M_{0} is a cocompact lattice in M_{0}, and Γ ∩ N_{0} is a cocompact lattice in N_{0}. Let 𝔞 denote the Lie algebra of A_{0}. We identify 𝔞 with its dual using the Killing form.

Let α_{1}, …, α_{r} denote the roots, which we view as elements of the dual of 𝔞. A Siegel set is a set 𝔖 = KMAN, where K is a maximal compact subgroup of G, M ∈ M_{0} and N ∈ N_{0} are compact, and A = {a ∈ A : α_{k}(log a) < C, 1 ≤ k ≤ r}, where C > 0 is some positive constant. It follows from reduction theory that for appropriate choices of M, N and C there exists a finite set J ∈ G such that for every g ∈ G, the intersection 𝔖 ∩ gΓJ is not empty.

Since G = KP_{0} = KM_{0}A_{0}N_{0} we may decompose g_{1} = k(g_{1})m_{0}(g_{1}) exp H(g_{1})n_{0}(g_{1}), where k(g_{1}) ∈ K, m_{0}(g_{1}) ∈ M_{0}, exp H(g_{1}) ∈ A_{0}, and n_{0}(g_{1}) ∈ N_{0}. Hence,

g_{1} = k(g_{1})m_{0}(g_{1}) exp H(g_{1})n_{0}(g_{1})  (23)

Then, for g_{1} ∈ 𝔖, we have for any root α_{j},

α_{j}(−H(g_{1})) > C  (24)

where C < 0 is an absolute constant. (Hence −H(g_{1}) is a finite distance from the positive Weyl chamber).

Let ρ_{k}, w_{k} be as in Condition A. Let P_{k} = M_{k}A_{k}N_{k} be the Langlands decomposition of P_{k}, where as above N_{k} ⊂ N_{0} is the unipotent radical of P_{k}, M_{k} ⊃ M_{0} is semisimple and A_{k} ⊂ A_{0} is one dimensional. Let d_{k}(g) = ∥ρ_{k}(gw_{k})∥. Then, by construction, if g = kman where k is in the maximal compact K, m ∈ M_{k}, a ∈ a_{k} and n ∈ N_{k}, then d_{k}(g) = d_{k}(a). It follows from structure theory that | log d_{k}(a) − c_{k}ω_{k}(log a)| < C_{1}, where C_{1} < 0 and c_{k} > 0 are absolute constants, and ω_{k} is the coroot corresponding to the root α_{k}; i.e. ω_{k}(α_{k}) = 1, and ω_{k}(α_{j}) = 0 if j ≠ k. Hence, if g = km_{0} exp(H(g))n_{0}, where k ∈ K, m_{0} ∈ M_{0}, a_{0} = exp(H(g)) ∈ A_{0}, n_{0} ∈ N_{0}, then log d_{k}(g) = ω_{k}(log a_{0}) = ω_{k}(H(g)).

Let

β_{k}(g) = max_{γ∈Γ} \frac{1}{d_{k}(gγ)^{1/c_{k}}}  (25)

It follows from reduction theory that up to an absolute constant, β_{k}(g) = β_{k}(g_{1}), where g_{1} is any element of gΓJ ∩ 𝔖. Hence,

| log β_{k}(g) − ω_{k}(−H(g_{1}))| < C  (26)

where H(g_{1}) is as in (23), and C is an absolute constant. In particular, it follows that the function β_{k} : G → ℝ is bounded from below away from 0.

We may apply the identity

α_{k} = ∑_{j}⟨α_{k}, α_{j}⟩ω_{j},  (27)

to H(g_{1}) and combine with (24) and (26) to obtain that for all g ∈ G and all k,

∑_{j}⟨α_{k}, α_{j}⟩ log β_{j}(g) > C′.  (28)

where C′ is an absolute constant.

Furthermore, for every constant C there exists a constant C′ such that if for some k and some g ∈ G and g_{1} ∈ gΓJ ∩ 𝔖 there exists g_{2} ∈ gΓJ such that \frac{1}{d_{k}(g_{2})} > \frac{1}{Cd_{k}(g_{1})}, d_{k}(g_{2}) ≠ d_{k}(g_{1}) then it follows from reduction theory that

α_{k}(−H(g_{1}))) < C′  (29)

(i.e. −H(g_{1}) is “near the k’th wall” of the positive Weyl chamber). Hence, in this case

∑_{j}⟨α_{k}, α_{j}⟩ log β_{j}(g) < C′′.  (30)

Choose q_{j} > 0 such that ∑_{j} q_{j}ω_{j} belongs to the positive Weyl chamber. Then for any k,

∑_{j} q_{j}⟨α_{j}, α_{k}⟩ > 0.  (31)

Let u_{j}(g) = β_{j}(g)^{1/q_{j}}, so that log β_{j}(g) = q_{j} log u_{j}(g). Hence, if (30) holds,

∑_{j}⟨α_{k}, α_{j}⟩q_{j} log u_{j}(g) < C′′  (32)

Since ⟨α_{k}, α_{k}⟩ > 0 and ⟨α_{k}, α_{j}⟩ ≤ 0 for j ≠ k, (32) may be rewritten as

u_{k} ≤ C ∏_{j≠k} u_{j}^{λ_{jk}},  (33)

where

λ_{jk} = \frac{q_{j}|⟨α_{j}, α_{k}⟩|}{q_{k}⟨α_{k}, α_{k}⟩}  (34)

Note that λ_{jk} ≥ 0 and in view of (31),

∑_{j≠k} λ_{jk} < 1.  (35)

We note that the functions u_{k} satisfy the estimate

u_{k}(g′g) ≤ ∥g′∥_{k}^{1/q_{k}}u_{k}(g)  (36)

where ∥ · ∥_{k} is the operator norm in the representation ρ_{k}.

Now choose δ > 0 such that (2) and Condition A hold for δ/q_{k} instead of δ, 1 ≤ k ≤ r. Let c and n be as in Condition A (with δ/q_{k} instead of δ). In view of (2) we may write μ^{(n)} = μ_{1} + μ_{2}, where μ_{1} has compact support and for any 1 ≤ k ≤ r,

∫_{G} ∥g∥_{k}^{δ} dμ_{2}(g) < \frac{1 − c}{3}.  (37)

Let A_{1} denote the averaging operator

(A_{1}f)(g) = ∫_{G} f(g′g) dμ_{1}(g′).

Let n > 0 be as in Condition A, and for sufficiently small δ > 0 and g ∈ G consider the average (A_{1}u_{k}^{δ})(g). If for every h in the support of μ_{1} the maximum in the definition of β_{k} (hence of u_{k}) is achieved by the same γ ∈ Γ, then, by Condition A we have, (A_{1}u_{k})(g) ≤ cu_{k}(g), with c < 1. If the maximum is achieved by different γ depending on the choice of g′ in the (compact) support of μ_{1}, then (29) holds. Hence, in that case (33) also holds. Thus in all cases, we obtain the system of inequalities:

A_{1}u_{k}^{δ} ≤ cu_{k}^{δ} + C′ ∏_{j≠k} u_{j}^{δλ_{jk}} + B′  (38)

where c < 1, C′ < ∞, B′ < ∞ and we have used again the compactness of the support of μ_{1}. The additive constant B′ arises because we have assumed that the u_{i} are bounded from below.

For ϵ > 0, (38) may be rewritten as

A_{1}(ϵu_{k})^{δ} ≤ c(ϵu_{k})^{δ} + ϵ_{1} ∏_{j≠k}(ϵu_{j})^{δλ_{jk}} + B′′

and in view of (35), ϵ_{1} > 0 can be made arbitrarily small by choosing ϵ > 0 small enough. Note that by (35), Jensen’s inequality and the fact that the u_{j} are bounded from below we have

∏_{j≠k}(ϵu_{j})^{δλ_{jk}} ≤ C_{1} ∑_{j≠k}(ϵu_{j})^{δ}.

Hence u = ∑_{k}(ϵu_{k})^{δ} satisfies the inequality

A_{1}u ≤ c′u + b,

where c′ < 1 − \frac{(1−c)}{3} and b > 0. Now in view of (37), we have

Au ≤ c_{1}u + b  (39)

where c_{1} < 1 and (Af)(g) = ∫_{G} f(g′g) dμ^{(n)}(g′).

4 Averaging operators

In this section we prove Proposition 2.7.

Lemma 4.1. Suppose H is a semisimple algebraic subgroup of GL(V) without compact factors, such that V does not have any H-invariant vectors. Suppose μ is a Zariski dense measure on H satisfying (2). Then, there exist N > 0 and c > 0 such that for all n > N and all nonzero v ∈ V,

\frac{1}{n} ∫_{H} log \frac{∥hv∥}{∥v∥} dμ^{(n)}(h) > c

Proof. See [Bou, Chapter III, Corollary 3.4 p. 53-54]. See also [Furm] and the original paper [Furs] for closely related statements.

Lemma 4.2. Let H, V and μ be as in Lemma 4.1, Then, for all sufficiently small δ > 0 there exist c_{1}, 0 < c_{1} < 1 and N > 0 such that for all n > N and all v ∈ V,

∫_{H} ∥hv∥^{−δ} dμ^{(n)}(h) < c_{1}∥v∥^{−δ}.  (40)

Proof. Without loss of generality, we assume that ∥v∥ = 1. We first show that there exists n > 0 for which(40) holds. Let n, c be as in Lemma 4.1. Since H is semisimple, H ⊂ SL(V). Hence there exists k > 0 depending only on the dimension of V, such that for h ∈ H, ∥h^{−1}∥ ≤ ∥h∥^{k}. Then, for all δ_{0} > 0,

∥hv∥^{−δ_{0}} ≤ ∥h^{−1}∥^{δ_{0}} ≤ ∥h∥^{kδ_{0}}.  (41)

Now, using Taylor’s theorem, for some 0 < δ_{0} < δ,

∥hv∥^{−δ} = 1 − δ log ∥hv∥ + \frac{1}{2}δ^{2}(log ∥hv∥)^{2}∥hv∥^{−δ_{0}}

Then, using (41),

∥hv∥^{−δ} ≤ 1 − δ log ∥hv∥ + \frac{1}{2}δ^{2}(log ∥h∥)^{2}∥h∥^{kδ}

Hence, for sufficiently small δ > 0, using Lemma 4.1,

∫_{H} ∥hv∥^{−δ} dμ^{(n)}(h) ≤ 1 − δnc + \frac{1}{2}δ^{2} ∫_{H}(log ∥h∥)^{2}∥h∥^{kδ} dμ^{(n)}(h)
≤ 1 − δnc + δ^{2}C_{n}

where C_{n} is independent of δ, and in the last line we have used the condition (2). Now choose δ < C_{n}/(2c). Then (40) holds for n. To see that it holds for m > n, note that μ^{(m)} = μ^{(n)} ∗ μ^{(m−n)} and μ^{(m−n)} is a probability measure.

Proof of Proposition 2.7. Let P_{k}, ρ_{k}, V_{k}, w_{k} be as in condition A. Let V_{k}^{H_{μ}} denote the H_{μ}-invariant subspace of V_{k} on which the action of H_{μ} is trivial, and let V′_{k} denote the complementary H_{μ}-invariant subspace of V_{k}. The assumption of Proposition 2.7 implies that Gw_{k} does not intersect V_{k}^{H_{μ}}. Let π′_{k} denote the projection onto V′_{k}. Let K denote the maximal compact subgroup of G. Since G = KP_{k}, and P_{k} stabilizes the span of w_{k}, Gw_{k} is projectively compact. Hence there exists a constant c_{0} > 0 such that for all v ∈ Gw_{k}, ∥π′_{k}(v)∥ > c_{0}∥v∥. Now the proposition follows from Lemma 4.2 applied to π′_{k}(v).

5 Proof of Lemma 2.2

Let X = G/Γ, and let σ be a locally finite μ-stationary measure on X. Consider the space G×X with the measure μ×σ. Then, since σ is μ-stationary, for any (compact) K ⊂ X,

(μ × σ){(g, x) : gx ∈ K} = σ(K).

Hence,

∫_{X} μ{ g : gx ∈ K} dσ(x) = σ(K)

Hence, for any compact subset C of X,

∫_{C} μ{ g : gx ∈ K} dσ(x) ≤ σ(K)

Now after convolving, we see that for any n > 0,

∫_{C} μ^{(n)}{ g : gx ∈ K} dσ(x) ≤ σ(K)

But by (R2), for n sufficiently large, for any x ∈ C, μ^{(n)}{ g : gx ∈ K} ≥ (1 − ϵ). Then,

σ(C) ≤ \frac{σ(K)}{1 − ϵ}

Since C is arbitrary, we see that σ(X) ≤ \frac{σ(K)}{1−ϵ}.

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