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Random Walks on Locally Homogeneous Spaces

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Random Walks on Locally Homogeneous Spaces RANDOM WALKS ON LOCALLY HOMOGENEOUS SPACES ALEX ESKIN AND ELON LINDENSTRAUSS Dedicated to the memory of Maryam Mirzakhani Abstract. Let G be real Lie group, and Γ a discrete subgroup of G. Let µ be a measure on G. Under a certain condition on µ, we classify the finite µ-stationary measures on G=Γ. We give an alternative argument (which bypasses the Local Limit Theorem) for some of the breakthrough results of Benoist and Quint in this area. Contents 1. Introduction1 2. General cocycle lemmas 15 3. The inert subspaces Ej(x) 30 4. Preliminary divergence estimates 37 5. The action of the cocycle on E 39 6. Bounded subspaces and synchronized exponents 44 7. Bilipshitz estimates 62 8. Conditional measures. 64 9. Equivalence relations on W + 67 10. The Eight Points 68 11. Case II 84 12. Proof of Theorem 1.2. 101 References 106 Index of Notation 109 1. Introduction 1.1. Random walks and stationary measures. Let G0 be a real algebraic Lie Group, let g0 denote the Lie algebra of G0, and let Γ0 be a discrete subgroup of G0. Let µ be a probablity measure on G0 with finite first moment. Let ν be an µ-stationary Research of the first author is partially supported by NSF grants DMS 1201422, DMS 1500702 and the Simons Foundation. Research of the second author is partially supported by the ISF (891/15). 1 2 ALEX ESKIN AND ELON LINDENSTRAUSS measure on G0=Γ0, i.e. Z µ ν = ν; where µ ν = gν dµ(g). ∗ ∗ G0 We assume ν(G0=Γ0) = 1, and also that ν is ergodic (i.e. is extremal among the µ- stationary measures). Let denote the support of µ, let G G0 denote the closure S Z S ⊂ of the group generated by , and let G GL(g0) denote the Zariski closure of the adjoint group Ad(G ). S S ⊂ S We say that a measure ν on G0=Γ0 is homogeneous if it is supported on a closed orbit of its stabilizer g G0 : g ν = ν . We recall the followingf 2 breakthrough∗ theoremg of Benoist-Quint [BQ1], [BQ2]: Theorem 1.1 (Benoist-Quint). Suppose µ is a compactly supported measure on G0, Z G is semisimple, Zariski connected and has no compact factors. Then, any ergodic S µ-stationary measure ν on G0=Γ0 is homogeneous. In this paper, we prove some generalizations and extensions of Theorem 1.1. For a discussion of related results, see 1.5. Recall that the measure µ hasx finite first moment if Z 1 log max(1; g ; g− ) dµ(g) < : G0 k k k k 1 One easy to state consequence of our results is the following: Theorem 1.2. Let G0 be a Lie group and let µ be a probability measure on G0 with Z finite first moment. Suppose G is generated by unipotents over C. Let Γ0 be a S discrete subgroup of G0, and let ν be any µ-stationary measure on G0=Γ0. Then, ν is G -invariant. S Another consequence is an alternative proof of an extension of Theorem 1.1 where the assumption that µ is compactly supported is replaced by the weaker assumption that µ has finite first moment. Thus, we prove the following: Z Theorem 1.3. Suppose µ is a measure on G0 with finite first moment, G is semisim- ple, Zariski connected and has no compact factors. Then, any ergodicSµ-stationary measure ν on G0=Γ0 is homogeneous. 1.2. The main theorems. Let µ(n) = µ µ µ (n times). If H is a Lie group, we denote the Lie algebra of H by Lie(H).∗ · · · ∗ G0 acts on g0 by the adjoint representation. For G0 = SL(n; R), g G0, v g, 2 2 1 Ad(g)v = gvg− : Notation. We will often use the shorthand (g) v for Ad(g)v. ∗ Let V be a vector space on which G0 acts. RANDOM WALKS ON LOCALLY HOMOGENEOUS SPACES 3 Definition 1.4. The measure µ is uniformly expanding on V if there exist C > 0 and N N such that for all v V , 2 2 Z g v (1.1) log k · k dµ(N)(g) > C > 0: v G k k Essentially, µ is uniformly expanding on V if every vector in V grows on average under the random walk. Lemma 1.5. µ is uniformly expanding on V if and only if for every v V , for N N 2 µ -a.e. ~g = (g ; g ; : : : ; g ;::: ) (G0) , 1 2 n 2 1 lim log (gn : : : g1) v > 0 n !1 n k · k Thus, if µ is uniformly expanding, then with probability 1 every vector grows exponentially. The proof of this lemma is postponed to 3. x Z Remark. We can consider the Adjoint action of G0 on its Lie algebra g0. If G is semisimple with no centralizer and no compact factors, then µ is uniformly expandingS Z on g0, see e.g. [EMar, Lemma 4.1]. However, there are many examples with G not S semisimple for which uniform expansion on g0 still holds, see e.g. (1.13) below. Instead of assuming that µ is uniformly expanding on g0 we consider the following somewhat more general setup, to accomodate centralizers and compact factors. Definition 1.6. Let Z be a connected Lie subgroup of G0. We say that µ is uniformly expanding mod Z if the following hold: (a) Z is normalized by G . (b) The conjugation actionS of G on Z factors through the action of a compact subgroup of Aut(Z). S (c) We have a G -invariant direct sum decomposition S g0 = Lie(Z) V; ⊕ and µ is uniformly expanding on V . (Note that V need not be a subalgebra). Our result is the following: Theorem 1.7. Let G0 be a real Lie group, and let Γ0 be a discrete subgroup of G0. Suppose µ is a probability measure on G0 with finite first moment, and suppose there exists a connected subgroup Z such that µ is uniformly expanding mod Z. Let ν be any ergodic µ-stationary probability measure on G0=Γ0. Then one of the following holds: (a) There exists a closed subgroup H G0 with dim(H) > 0 and an H-homogeneous ⊂ probability measure ν0 on G0=Γ0 such that the unipotent elements of H act er- godically on ν0, and there exists a finite µ-stationary measure λ on G0=H such 4 ALEX ESKIN AND ELON LINDENSTRAUSS that Z (1.2) ν = gν0 dλ(g): G Let H0 denote the connected component of H containing the identity. If 0 0 0 dim H is maximal then H and ν0 are unique up to conjugation of H and the obvious corresponding modification of ν0. (b) The measure ν is G -invariant and is supported on a finite union of compact subsets of Z-orbits. S In partucular, if µ is uniformly expanding on g0 we may take Z = e , and alter- native (b) says that ν is G -invariant and finitely supported. f g Note that the subgroup SH of Theorem 1.7(a) may not be connected. Definition 1.8. A real-algebraic Lie subgroup L G0 is called an H-envelope if the following hold: ⊂ (i) L H and H0 is normal in L. (ii) The⊃ image of H in L=H0 is discrete. (iii) There exists a representation ρ0 : G0 GL(W ) and a vector v W such ! L 2 that the stabilizer of vL is L. 0 0 Suppose H is as in Theorem 1.7. Let NG0 (H ) denote the normalizer of H in G0. Vdim H Let ρH (g0) denote the vector v1 vn, where v1;::: vn is a basis of 0 2 ^ · · · ^ Lie(H ). Then the stabilizer of ρH in G0 is an H-envelope. Also, if G0 is an algebraic 0 group and Γ0 is an arithmetic lattice, the Zariski closure in G0 of Γ0 NG0 (H ) is an H-envelope. \ The following is an easy consequence of Theorem 1.7: Theorem 1.9. Let G0 be a real Lie group, and let Γ0 be a discrete subgroup of G0. Suppose µ is a probability measure on G0 with finite first moment. Let ν be any ergodic µ-stationary probability measure on G0=Γ0 such that (a) of Theorem 1.7 holds, let H, ν , λ be as in Theorem 1.7. Suppose L G0 is an H-envelope. Then 0 ⊂ (a) There exists a finite µ-stationary measure λ~ on G0=L (the image of λ under the natural projection G0=H G0=L). (b) Suppose G L. Then either! ν is supported on finitely many H0-orbits, or there existsS a⊂ vector v Lie(L) such that Z2 1 (n) (1.3) lim sup log (Ad(g)v) ρH dµ (g) 0: n n G0 k ^ k ≤ !1 Vdim H Here, as above, ρH (g) is the vector v1 vn, where v1;::: vn is a basis of Lie(H0). 2 ^ · · · ^ Proof of Theorem 1.9. Since the projection G0=H G0=L commutes with the ! left action of G0, the image of any µ-stationary measure on G0=H is a µ-stationary RANDOM WALKS ON LOCALLY HOMOGENEOUS SPACES 5 measure on G0=L. Thus (a) holds. Now (b) follows by applying Theorem 1.7 to the random walk (with measure µ) on L=H = (L=H0)=(H=H0). Remark. Under the assumptions of Theorem 1.9, there exists a representation G0 GL(W ) and a vector v W such that the stabilizer of v is L.
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