Character rigidity for lattices and commensurators I after Creutz-Peterson
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1 Character rigidity for lattices ad commesurators I after Creutz-Peterso Talk C3 for the Arbeitsgemeischaft o Superridigity held i MFO Oberwolfach, 31st March - 4th April Sve Raum 1 Itroductio The aim of talks C3 ad C4 is to give a proof of the followig theorem. Theorem 1 ([CP13]). Let G be a simple o-compact Lie group with property (T) ad trivial cetre ad let H be a product of p-adic Lie groups. Let Λ be a irreducible lattice i G H. The every extremal character τ Λ C is a is either almost periodic or τ = δ e is the left-regular character. Here a locally compact group G has the Howe-Moore property if ay uitary represetatio of G either has a fixed vector or it is mixig. Furthermore, a lattice Γ G 1 G is called irreducible, if the projectios π i (Γ) G i are dese for all i {1,..., }. Let us metio a importat example of groups satisfyig the hypothesis of the previous theorem. Example 2. The diagoal embeddig of PSL(, Z[ 1 p ]) ito PSL(, R) PSL(, Q p) is a irreducible lattice for all prime umbers p ad all 2. If 3, the PSL(, R) ad PSL(, Q p ) satisfy all other hypotheses of Theorem 1. We will deduce Theorem 1 o lattices i G H from the ext theorem about dese subgroups of G. Recall that the Schlichtig completio of a iclusio Γ Λ by defiitio is the closure of the image Λ Sym(Λ/Γ) i the topology of poitwise covergece. If Γ is commesurated by Λ, the the closure of Γ i Λ//Γ is a compact ope subgroup. Theorem 3 (Theorem D of [CP13]). Let G be a simple o-compact Lie group with property (T) ad trivial cetre. Assume that Λ is a coutable dese subgroup of G which cotais ad commesurates a lattice Γ of G ad such that the Schlichtig completio Λ//Γ is a product of topologically simple groups with the Howe-Moore property. The every extremal character τ Λ C is either almost periodic or τ = δ e. Proof that Theorem 3 implies Theorem 1. Let Λ G H be a irreducible lattice as i the statemet of Theorem 1. Sice H is a product of p-adic Lie groups, it is totally discoected. Hece there is a compact ope subgroup U H. Note that every compact ope subgroup of H is commesurated by H. Let Γ = Λ (G U). The Γ is commesurated by Λ. Moreover, usig the fact that the projectio of Λ to H is dese i H ad the fact that U is compact ope i H, the Schlichtig completio Λ//Γ satisfies Λ//Γ H//U H. 1 last modified April 8,
2 Deote by π G H G the projectio oto the first factor. Sice Λ is irreducible i G H it follows that π(λ) is dese i G. Furthermore, π(γ) is discrete i G, sice U is compact. It follows that π(γ) π(λ) G satisfies all hypotheses of Theorem 3. To coclude the proof, we remark that by Margulis s ormal subgroup theorem every ormal subgroup of Γ has fiite idex, so that π is ijective o Γ. Example 4. Let us cotiue Example 2. Cosider the irreducible lattice Λ = PSL(, Z[ 1 p ]) i G H = PSL(, R) PSL(, Q p ). Sice PSL(, Z p ) is compact ope i PSL(, Q p ), we have to cosider Γ = PSL(, Z[ 1 p ]) PSL(, R) PSL(, Z p) = PSL(, Z) i Theorem 3. Ideed, oe ca check that PSL(, Z[ 1 p ])//PSL(, Z) PSL(, Q p) ad the latter group is simple ad has the Howe- Moore property. We preset ow a vo Neuma algebraic versio of Theorem 3. This is the statemet that we are goig to prove. Thaks to the correspodece betwee extremal characters o Λ ad fiite factor represetatios of Λ, which was preseted i the earlier talk C1, it suffices to show the follow theorem. Theorem 5 (Theorem B of [CP13]). Let G be a simple o-compact Lie group with property (T) ad trivial cetre. Assume that Λ is a coutable dese subgroup of G which cotais ad commesurates a lattice Γ of G ad such that the Schlichtig completio Λ//Γ is a product of topologically simple groups with the Howe-Moore property. If π Λ U(M) is a fiite factor represetatio of Λ such that π(λ) = M, the either M is fiite dimesioal, or π exteds to a isomorphism L(Λ) M. The proof of this theorem splits ito two parts. I the ext lecture C4, we are goig to see that uder the hypothesis of Theorem 5, either π exteds to a isomorphism L(Λ) M or π(γ) is a ameable vo Neuma algebra. The latter assumptio is the startig poit of this lecture. For the proof of Theorem 5 we are goig to assume that π(γ) is ameable. I particular, π caot exted to to a isomorphism L(Λ) M, sice Γ is a property (T) group. (See Sectio 3). 2 Proof of the mai theorem We proceed i a direct way to the proof of Theorem 5 (uder the assumptio that π(γ) is ameable). Three results are goig to be used i the proof. They are further explaied i Sectio 4 ad 3, respectively. Propositio 6. Let Γ be discrete coutable group with property (T) ad let π Γ M be a odegeerate represetatio ito a fiite vo Neuma algebra. The M has property (T). Propositio 7. Let M be a fiite vo Neuma algebra which is ameable ad has property (T). The M is discrete. Theorem 8. Let G be a o-discrete totally discoected simple group with the Howe-Moore property. The there is o o-trivial cotiuous homomorphism of G ito the uitary group of a fiite vo Neuma algebra. 2
3 We give a proof of our mai Theorem 5 assumig these two results ad prove them afterwards. Proof of Theorem 5 uder the assumptio that π(γ) is ameable. Let π Λ U(M) be a fiite factor represetatio of Λ such that π(λ) = M ad π(γ) is ameable. Sice Γ is a lattice i a property (T) group, it has property (T) itself. By Propositio 6, it follows that π(γ) is a fiite ameable vo Neuma algbera with property (T). So Propositio 7 shows that π(γ) is completely atomic. Cosider the represetatio π π op Λ U(L 2 (M) L 2 (M)), which is defied by (π π op )(λ) = π(λ) Jπ(λ)J. Sice π(γ) is completely atomic, there is a o-zero (π π op )(Γ)-ivariat vector ξ L 2 (M) L 2 (M). To see this, it suffices to check that if e 1,..., e deotes a orthoormal basis of C, the i=1 e i e i is ivariat uder all operators U U with U U(). Ideed, i=1 Ue i U e i = i=1 k,l=1 u ki u il e k e l = δ k,l e k e l = e k e k. k,l=1 k=1 We show that ξ is (π π op )(Λ)-ivariat. Deote by p the projectio oto the closed liear spa of (π π op )(Λ)ξ. Sice p(π π op )(Λ)p = (π π op )(Λ)p, it follows that p (π π op )(Λ). Defie M = (π π op )(Λ) p ad ote that M M M is a fiite vo Neuma algebra. A sequece (λ ) i Λ goes to e i Λ//Γ if ad oly if for every fiite idex subgroup Γ 0 of Γ there is N N such that λ Γ 0 for all N. It follows that if (λ ) is a sequece i Λ that goes to e i Λ//Γ, the (π π op )(λ λ)ξ (π π op )(λγ)ξ for big eough. Cosequetly, (π π op )(λ )p p strogly. So (π π op )p exteds to a strogly cotiuous represetatio of Λ//Γ ito the fiite vo Neuma algebra M. By Theorem 8 we ifer that this represetatio is trivial. I particular, (π π op )(λ)ξ = (π π op )(λ)pξ = pξ = ξ for all λ Λ. It follows that the vo Neuma algebra (π π op )(Λ) is ot diffuse. So either is M = π(λ). Sice M is a factor, it follows that M is fiite dimesioal. 3 A rigidity vs. ameability result I this sectio we show that property (T) ad ameability are icompatible ot oly o the level of groups but also o the level of vo Neuma algebras. I order to illustrate clealy the strategy of the proof, we make use of the famous equivalece of ameability ad hyperfiiteess of vo Neuma algebras [Co76]. This is absolutely ot ecessary, but slightly simplifies the argumet. Defiitio 9 (Property (T) for fiite vo Neuma algebras). Let M be a fiite vo Neuma algebra with faithful tracial state τ. We say that M has property (T), if the followig coditio holds. Wheever (Ψ ) is a sequece of uital completely positive ad trace preservig maps such that Ψ (x) x 2 0 as for all x M, the Ψ id uiformly i 2 o (M) 1. Propositio 6 (Recall). Let Γ be discrete coutable group with property (T) ad let π Γ M be a o-degeerate represetatio ito a fiite vo Neuma algebra. The M has property (T). Proof. Deote by τ a faithful ormal trace o M. Let (Ψ ) be a sequece of uital completely positive maps o M that coverges to id M poitwise i 2. The fuctios ϕ (g) = τ(ψ (π(g))π(g) ) = Ψ π(g)jπ(g)j1, 1 L 2 (M) 3
4 are of positive type. Note that ϕ 1 poitwise. Sice Γ has property (T), it follows that ϕ 1 uiformly. So Ψ (π(g)) π(g) 2 2 = Ψ (π(g)) 2 + π(g) 2 2 2Re Ψ (π(g), π(g) = 2 2Reϕ (g) 0 uiformly for all g Γ. This implies that Ψ (x) x 2 0 uiformly o the 2 -closure C of π((cγ) L 1 (Γ),1). By Kaplasky s desity theorem, C = M. So it follows that C cotais a ope set. We ifer that (Ψ ) coverges uiformly i 2 to id M o (M) 1. Propositio 7 (Recall). Let M be a fiite vo Neuma algebra which is ameable ad has property (T). The M is discrete. Proof. Assume that M is ot discrete uder the hypothesis of the propositio. The there is a cetral projectio p Z(M) such that pm is diffuse. Sice pm is ameable ad has property (T), we may replace 1 M by p ad assume that M is diffuse. Sice M is ameable, there is a sequece of fiite rak completely positive uital maps Ψ M M such that Ψ (x) x i 2 as. Sice M is diffuse there is a sequece of uitaries (u ) i M such that u 0 weakly as. By property (T), there is some N N such that Ψ N (u ) u 2 2 < 1/2 for all N. Deote by A the fiite dimesioal image of Ψ N. Because (u ) goes to 0 weakly, there is some M N such that τ(u M x) < 1/4 for all x A. But this implies Ψ N (u M ) u M 2 2 = Ψ N (u M ) u M 2 2 2Reτ(Ψ(u M)u M ) > = 1 2, cotradictig Ψ N (u ) u 2 2 < 1/2. We have show that M caot be diffuse, which fiishes the proof. 4 Absece of fiite vo Neuma algebra represetatios of simple Howe-Moore groups Theorem 8 (Recall). Let G be a o-discrete totally discoected simple group with the Howe- Moore property. The there is o o-trivial cotiuous homomorphism of G ito the uitary group of a fiite vo Neuma algebra. Proof. Let π G U(M) be a o-degeerate represetatio of G ito a fiite vo Neuma algebra. Let z Z(M) be the maximal cetral projectio such that π(g) z = zc1. Replacig M by (1 z)m, we have to show that there is a o-trivial cetral projectio i Z(M) uder which π is trivial. Sice G is totally discoected the space {ξ L 2 (M) ξ is K-ivariat for some K G compact ope} is dese i L 2 (M). Hece there is a compact ope subgroup K G ad a K-ivariat vector ξ L 2 (M) such that ξ ˆ1 2 < 1/4. It follows that the positive type fuctio ϕ g π(g)ξ, ξ satisfies ϕ(gh) ϕ(hg) < 1/2. Moreover, ϕ is idetically 1 o K. 4
5 Sice G is simple ad K is commesurated by G, [BH89, Theorem 3] implies that the set {[K K gkg 1 ] g G} is ubouded. If follows that g G gkg 1 ca ot be covered by fiitely may cosets of K ad it is hece ot compact. Sice ϕ( g G gkg 1 ) (1/2, 1], the represetatio π is ot mixig. Usig the Howe-Moore property, we ifer that there is a o-zero G-fixed vector ξ for π. Now let p B(L 2 (M)) be the projectio oto the G-ivariat vectors. We already saw that p 0. Moreover, p is ivariat uder multiplicatio by elemets from M = π(g) ad M = π(g). So p Z(M). Sice π(g)p = p for all g G, we foud a o-zero cetral projectio i M uder which π is trivial. This fiishes the proof. Ackowledgemets We wat to thak Fracois Le Maître for suggestig several improvemets of these lecture otes. Refereces [BH89] G. M. Bergma ad L. W. Hedrik. Subgroups close to ormal subgroups. J. Algebra 127 (1), 80 97, [Co76] A. Coes. Classificatio of ijective factors. Cases II 1, II, III λ, λ 1. A. Math. (2) 74, , [CP13] D. Creutz ad J. Peterso. Character rigidity for lattices ad commesurators. arxiv:
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