THE ORNSTEIN-WEISS LEMMA FOR DISCRETE AMENABLE GROUPS.

Transcription

1 THE ORNSTEIN-WEISS LEMMA FOR DISCRETE AMENABLE GROUPS FABRICE KRIEGER Abstract In ths note we prove a convergence theorem for nvarant subaddtve functons defned on the fnte subsets of a dscrete amenable group The theorem can be proved usng a quas-tlng result due to DS Ornsten and B Wess but the proof gven here follows deas of M Gromov Let us pont out that we generalze here to dscrete amenable groups the verson of the convergence theorem gven for countable amenable groups n a precedng paper by the same author 1 Introducton Let G be a group We denote by F(G) the set of all fnte subsets of G Let K and A be subsets of G The K-boundary of A denoted by K (A) s the set of all elements g n G such that Kg = {kg : k K} ntersects both A and G \ A There are several equvalent defntons of amenable groups n the lterature The followng s a characterzaton due to Følner [Føl] For a more complete descrpton of ths class of groups see for example [Gre] or [Pat] A (dscrete) group G s sad to be amenable f for all ɛ > 0 and for all K F(G), there exsts F F(G) satsfyng K (F ) ɛ F, where F denotes the cardnalty of the set F Such a set F s called an (ɛ, K)- nvarant set of G, or an (ɛ, K)-nvarant Følner set of G It can be shown that a group G s amenable f and only f there exsts a net (F ) I of elements of F(G) such that lm K (F ) F for every K F(G) Such a net (F ) I s called a Følner net of G The class of amenable groups ncludes fnte groups, abelan groups, t s closed under the operatons of takng subgroups, takng factors, takng extensons and takng ncreasng unons Typcal examples of non-amenable groups are the groups contanng a subgroup somorphc to a non-abelan free group But ths property of contanng a non-abelan free group don t characterze at all the non-amenable groups (see for example [OlS]) The am of ths paper s to prove the followng convergence theorem: Theorem 11 (Ornsten-Wess) Let G be an amenable group and h: F(G) R a functon satsfyng the followng condtons: 1991 Mathematcs Subject Classfcaton 43A07, 20F65, 20F19, 28C10 Key words and phrases Amenable group, subaddtve functon, Ornsten-Wess Lemma, entropy, mean topologcal dmenson 1 = 0

2 2 FABRICE KRIEGER (a) h s subaddtve, e (b) h s rght nvarant, e h(a B) h(a) + h(b) for all A, B F(G); h(ag) = h(a) for all g G and A F(G) Then, for every Følner net (F ) I for G, the lmt λ = λ(g, h) = lm h(f ) F exsts and s fnte Moreover, ths lmt does not depend on the choce of the Følner net for G Ths theorem s a generalzed verson of the one contanng n [Kr] For countable amenable groups and wth more stronger condtons of h, ths convergence theorem was proved usng the Ornsten-Wess quas-tlng result [OrW, Secton I2, Th 6] n [LW, Th 61] In [Gro, Secton 13], Gromov gves a sketch of the proof of 11 by usng tools ntroduced n [OrW] The proof gven here follows the deas of Gromov Theorem 11 s used to defne topologcal nvarants of amenable group actons as metrc entropy, mean topologcal dmenson (see [Gro], [LW], [OrW], [CoK]) One can fnd n [Mou] such a convergence theorem for nvarant functons satsfyng a more stronger assumpton than subaddtvty The result n [Mou] s suffcent for defnng the metrc entropy of amenable groups The paper s organzed as follows In Secton 2, we recall the noton of K- boundary of a subset of a group G and the defnton of Følner subsets Then we establsh the Følner characterzaton of amenablty n terms of Følner nets for (non-necessary countable) dscrete groups We prove n Secton 3 the Fllng-lemma (Lemma 35) used n the nducton step of the proof of Theorem 11 In Secton 4, we prove the Theorem 11 The dea of the proof s the followng We construct by nducton a process needed to ɛ-cover every large Følner subset D of the group by translates of some fxed small Følner subsets The property of ths partcular coverng wll make easy the estmaton of h(d) needed n the proof of the convergence theorem Ths work was done at Max Planck Insttute for Mathematcs (Bonn, Germany) The author s grateful for the fnancal support and hosptalty of the Insttute durng hs stay (Aprl 2010) 2 Amenablty In ths secton we ntroduce the notons of K-nteror, K-exteror and K-boundary of a subset of a group G These tools are very convenent for provng the Fllnglemma (Lemma 35) We present the Følner characterzaton of amenablty usng K-boundares We ntroduce Følner nets whch are generalzatons of Følner sequences and are very convenent for handlng wth non-countable amenable groups 21 Relatve amenablty Let K and A subsets of a group G The K-nteror (resp K-exteror) of A s the subset Int K (A) (resp Ext K (A)) of the elements g n G such that Kg = {kg : k K} s contaned n A (resp n G \ A) We defne the K-boundary of A as follow: K (A) = G \ ( Int K (A) Ext K (A) )

3 ORNSTEIN-WEISS LEMMA 3 Thus, the K-boundary of A s the subset of all elements g n G such that Kg ntersects both A and G \ A An mmedate consequence of the defnton of K-boundary s the followng: Proposton 21 Let K, A, B be subsets of a group G and g an element of G We have: () K (A) = K (G \ A) ; () K (A B) K (A) K (B) ; () K (A \ B) K (A) K (B) ; (v) K (A) K (A) s K K G ; (v) Kg (A) = g 1 K (A) ; (v) K (Ag) = K (A)g Suppose K and A be fnte subsets of G Then K (A) s fnte Suppose A We defne the relatve amenablty constant of A wth respect to K denoted by α(a, K) by: α(a, K) = K(A) A Equaltes (v) et (v) of Proposton 21 mply (21) α(a, Kg) = α(ag, K) = α(a, K) for all g G If α = α(a, K), the set A s called an (α, K)-nvarant Følner subset of G or smply an (α, K)-nvarant subset of G Recall that a dscrete group G s sad to be amenable f for all ɛ > 0 and K F(G) there exsts an (ɛ, K)-nvarant subset of G If the group G s countable then the amenablty of G s equvalent to the exstence of a Følner sequence (F n ) of G, e a sequence of elements of F(G) satsfyng the followng: K (F n ) lm = 0 for all K F(G) n F n If the group s not countable, there s an analogue of Følner sequences n terms of nets (see Proposton 22 below) Nets are very useful tools snce much results about sequences n topologcal spaces extend to nets Let us recall some results about nets needed n ths paper 22 Basc facts about nets We gve here some basc defntons and results about nets (also called generalzed sequences) For more detals, see for example [Ke] or [DuS] Recall that a partally ordered set (I, ) s sad to be drected f I s not empty and f every fnte subset of I has an upper bound A map f from a drected set I to a set X s called a net n X We wll use the notaton x nstead of f() (for I) and also (x ) nstead of f A net (x ) n a topologcal space X s sad to converge to x X f for every open neghborhood V of x, there exsts 0 I such that x V for all 0 If (x ) converges to x, we note lm x = x Let (x ) and (y j ) be nets n a topologcal space X The net (y j ) s called a subnet of (x ) f there exsts a functon ϕ: J I satsfyng the two condtons: (1) y j = x ϕ(j) for all j J; (2) for all I there s m J wth the property that, f j m then ϕ(j)

4 4 FABRICE KRIEGER The set C of all cluster ponts (sometmes called lmt ponts) of a net (x ) n a topologcal space X s the (closed) subset of X defned by C = I {x k : k } The pont x s a cluster pont of the net (x ) f and only f there s a subnet (y j ) of (x ) convergng to x If C, the lmt nferor lm nf x (resp lmt superor lm sup x ) of a net (x ) of real numbers s the supremum (resp the nfmum) of ts cluster ponts If lm nf x (resp lm sup x ) s fnte, then t s the mnmum (resp maxmum) of the set of cluster ponts Thus, f lm nf x = lm sup x < then the net (x ) wll converge to ts unque cluster pont Recall also that every net n a compact space admts at least one cluster pont 23 Amenablty and Følner nets The next result gves a characterzaton of amenable dscrete groups n terms of Følner nets: Proposton 22 A (dscrete) group G s amenable f and only f there exsts a net (F ) I n F(G) satsfyng lm K (F ) F = 0, for all K F(G) Such a net (F ) s called a Følner net of G Proof Suppose that G s amenable Let I be the (non-empty) set defned by Drect I as follow: I = {(ɛ, K): ɛ > 0 and K F(G)} (ɛ 2, K 2 ) (ɛ 1, K 1 ) ɛ 2 ɛ 1 and K 1 K 2 As G s amenable, we can choose for every = (η, L) an (η, L)-nvarant subset F of G Ths defne a net (F ) n F(G) Let ɛ > 0 and K F(G) Let 0 = (ɛ, K) For = (η, K ) 0 we have K (F ) F K (F ) F η ɛ, snce K K and snce η ɛ (see Proposton 21(v)) Hence lm K (F ) F = 0 Suppose now that there exsts a net (F ) of fnte subsets of G satsfyng lm K (F ) F for all K F(G) We wll show that G s amenable In fact, let ɛ > 0 and K F(G) As lm K (F ) F = 0, there s an 0 I such that K(F ) F ɛ for all 0 In partcular, F 0 s (ɛ, K)-nvarant Hence G s amenable = 0, 3 The fllng lemma In ths secton we ntroduce some tools needed for provng the Fllng-lemma (Lemma 35) Ths lemma s the key-result n the nducton step of the proof of Theorem 11

5 ORNSTEIN-WEISS LEMMA 5 Let X be a set and ɛ > 0 A famly (A ) I of fnte subsets of X s sad to be ɛ-dsjont f there s a famly (B ) I of dsjont subsets of X such that B A and B (1 ɛ) A for all I Lemma 31 Let X be a set and (A 1, A 2,, A n ) be an ɛ-dsjont famly of subsets of X Then (1 ɛ) A A Proof Snce (A 1, A 2,, A n ) s ɛ-dsjont, there exsts a dsjont famly (B 1, B 2,, B n ) of subsets of X such that B A and B (1 ɛ) A for all 1 n Thus (1 ɛ) A B = B A Lemma 32 Let G be a group, K a fnte subset of G, and 0 < ε < 1 Let A 1, A 2,, A n be an ɛ-dsjont famly of non empty fnte subsets of G and let η > 0 such that α(a, K) η for all 1 n Then one has α( A, K) η 1 ɛ Proof Usng Proposton 21(), we obtan K ( A ) K (A ) Thus K ( A ) K (A ) = α(a, K) A η As the famly (A ) 1 n s ɛ-dsjont, Lemma 31 mples (1 ɛ) A A We deduce α( A, K) = K( n A ) n A η 1 ɛ A Lemma 33 Let G be a group and let K, A and Ω be fnte subsets of G such that A Ω Suppose that there exsts ɛ > 0 such that Ω \ A ɛ Ω Then α(ω \ A, K) Proof The Proposton 21() gves α(ω, K) + α(a, K) ɛ K (Ω \ A) K (Ω) K (A) Thus K (Ω \ A) K (Ω) + K (A) = α(ω, K) Ω + α(a, K) A Snce Ω \ A ɛ Ω ɛ A, we deduce α(ω \ A, K) = K(Ω \ A) Ω \ A α(ω, K) + α(a, K) ɛ

6 6 FABRICE KRIEGER Lemma 34 Let G be a group and let A and B be two fnte subsets of G Then one has Ag B = A B g G Proof For E G, denote by χ E : G {0, 1} the characterstc foncton of E We have Ag B = χ Ag B (g ) = χ A (g g 1 )χ B (g ) g G g G g G g G g G Now, after changng the order of summaton and changng the varable, we obtan: Ag B = χ A (g g 1 ) = B A g G g G χ B (g ) g G Let G be a group Let K and Ω be fnte subsets of G and ɛ > 0 A subset R G s called an (ɛ, K)-fllng of Ω f the followng condtons are satsfed: (C1) R Int K (Ω); (C2) the famly (Kg) g R s ɛ-dsjont Remark that an (ɛ, K)-fllng s a fnte set and that t could be empty Lemma 35 (Fllng-lemma) Let Ω and K be non-empty fnte subsets of a group G For all ɛ ]0; 1], there exsts a fnte subset R G such that: (a) R s an (ɛ, K)-fllng of Ω; (b) g R Kg ɛ(1 α0 ) Ω, where α 0 = α(ω, K) s the relatve amenablty constant of Ω wth respect to K Proof Snce K, we can suppose 1 G K (otherwse choose k 0 K, replace K wth Kk0 1 and remark that α(ω, K) = α(ω, Kk0 1 ) accordng to Equaltes (21)) As 1 G K, we have Int K (Ω) Ω and Ext K (Ω) G \ Ω We deduce thus (31) Ω \ K (Ω) = Int K (Ω) (1 α 0 ) Ω Ω \ K (Ω) = Int K (Ω) Snce Int K (Ω) Ω, every (ɛ, K)-fllng of Ω s contaned n Ω and has a bounded cardnalty Thus we can choose an (ɛ, K)-fllng R G of Ω wth maxmal cardnalty Defne A = g R Kg We wll prove that A ɛ(1 α 0) Ω, whch s exactly condton (b) Lemma 34 mples (32) Kg A K A Let us prove that (33) g Int K (Ω) ɛ K Kg A for all g Int K (Ω) If g R, then Kg A = Kg and (33) s true snce ɛ 1 Int K (Ω) \ R and Kg A < ɛ K Then Kg \ A > (1 ɛ) Kg, Suppose now g

7 ORNSTEIN-WEISS LEMMA 7 whch mples that R {g} s an (ɛ, K)-fllng of Ω Ths contradcts the maxmalty of the cardnalty of R Thus Inequalty (33) s true We deduce (34) ɛ K Int K (Ω) Kg A Inequaltes (31), (32) and (34) mply Let us frst gve some remarks: g Int K (Ω) A ɛ(1 α 0 ) Ω 4 Proof of the Theorem 11 (1) If one choose A = B n condton (a) of Theorem 11, we get h(a) 2h(A) for all A F(G) Ths shows that h 0 (2) To prove Theorem 11 t s suffcent to prove the exstence of the lmt of (h(f )/ F ) for all Følner net (F ) of G In fact, the lmt wll be ndependent of the choce of the Følner net To see ths fact, let (A ) and (B j ) be two Følner nets of G Let Î = {î: I} (resp Ĵ = {ĵ : j J}) a copy of the drected set I (resp J) Drect the set K = (I J) (Î Ĵ) wth the bnary relaton defned n the natural way by: (, j) (, j ) and j j f (, j) I J and, j I J, (, j) (î, ĵ ) and j j f (, j) I J and î, ĵ Î Ĵ, (î, ĵ) (, j ) and j j f (î, ĵ) Î Ĵ and (, j ) I J, (î, ĵ) (î, ĵ ) and j j f (î, ĵ) Î Ĵ and (î, ĵ ) Î Ĵ Now defne the net (F k ) as follows: f k = (, j) then let F k = A and f k = (î, ĵ) then let F k = B j Defned n ths way, the net (F k ) s a Følner net of G Moreover, f ( h(f k )/ F k ) converges to λ wth respect to the drected set K, then both nets ( h(a )/ A ) and ( h(b j )/ B j ) converge to λ wth respect to ther drected set Proof of Theorem 11 Let (F ) be a Følner net of G and fx ɛ ]0, 1 2 ] Remark that the propertes of h mply that h(a) h({1 G }) A for all A F(G) whch shows that the net defned by x = h(f) F s bounded More precsely, the numbers x are contaned n [0, h({1 G })] As every net n a compact space admts at least one cluster pont, we can defne the real number λ = lm nf x, whch s n fact the least cluster pont Fx an nteger n 2 Then there exsts a fnte sequence K 1, K 2,, K n extracted from (F ) and satsfyng the followng condtons: (C1) h(k j )/ K j λ + ɛ for all 1 j n, (C2) α(k j, K ) ɛ 2n for all 1 < j n In fact, as λ s the least cluster pont of x we can fnd a subnet (x ϕ(k) ) k K and k 0 K satsfyng x ϕ(k) λ + ɛ, for all k k 0 Remark that (F ϕ(k) ) s also a Følner net of G, e for all K F(G) we have lm k K (F ϕ(k) ) F ϕ(k) = 0 Thus, t s possble to extract a fnte sequence

8 8 FABRICE KRIEGER K 1, K 2,, K n from (F ϕ(k) ) satsfyng condton (C2) (41) Let D be a non-empty fnte subset of G such that α(d, K j ) ɛ 2n for all 1 j n We wll show that for a large enough nteger n, there s an ɛ-dsjont famly n D composed by certan translates of the type K j g (wth 1 j n and g G) whch partally cover D, e such that the proporton of D covered be these sets s at least 1 ɛ After that, we wll use ths partal cover and the propertes of h to prove lm sup h(f )/ F λ, endng the proof of the Theorem 11 Let us defne by nducton a process to ɛ-cover D n at most n steps: Step 1 Recall that α(d, K j ) ɛ 2n for all 1 j n Usng Lemma 35 wth Ω = D and K = K n, there s R n G an (ɛ, K n )-fllng of D such that g R n K n g ɛ ( 1 α(d, K n ) ) ɛ(1 ɛ 2n ) Put D 1 = D \ g R n K n g The prevous nequalty mples: (42) D 1 ( 1 ɛ(1 ɛ 2n ) ) We contnue ths coverng process by nducton as follows Put D 0 = D Suppose that the coverng process apples k tmes, wth 1 k n 1 The nducton hypothess at step k s: (H1) α(d k 1, K j ) (2(k 1) + 1)ɛ 2n k+1 for all 1 j n k + 1; (H2) R n k+1 G s an (ɛ, K n k+1 )-fllng of D k 1 ; (H3) If we wrte D k = D k 1 \ K n k+1 g, g R n k+1 then D k k 1 =0 ( 1 ɛ ( 1 (2 + 1)ɛ 2n )) Remark that ths hypothess s satsfed for k = 1 Let us construct step k + 1 : Step k + 1 If D k ɛ D k 1 then D k ɛ and we stop the coverng process Otherwse, we have D k > ɛ D k 1 Let 1 j n k Lemma 33 mples (43) α(d k, K j ) α( g R n k+1 K n k+1 g, K j ) ɛ Equaltes (21) and condton (C2) mply α(k n k+1 g, K j ) = α(k n k+1, K j ) ɛ 2n + α(d k 1, K j ) ɛ Snce the famly (K n k+1 g) g Rn k+1 s ɛ-dsjont, Lemma 32 gves α( g R n k+1 K n k+1 g, K j ) ɛ2n 1 ɛ

9 ORNSTEIN-WEISS LEMMA 9 Usng Inequalty (43) and the nducton hypothess (H1), we deduce ( ) α(d k, K j ) ɛ2n 2(k 1) + 1 ɛ 2n k+1 (1 ɛ) ɛ + (2k + 1)ɛ 2n k ɛ for all 1 j n k The latter nequalty s (H1) for k + 1 Usng Lemma 35 wth Ω = D k and K = K n k, we get the exstence of R n k G an (ɛ, K n k )-fllng of D k satsfyng g R n k K n k g ɛ ( 1 α(d k, K n k ) ) ɛ ( 1 (2k + 1)ɛ 2n k) D k In partcular, hypothess (H2) s satsfed for k + 1 Defne D k+1 = D k \ K n k g g R n k Then we have D k+1 D k ( 1 ɛ ( 1 (2k + 1)ɛ 2n k)) Usng the nducton hypothess (H3) and the latter nequalty, we obtan D k+1 k ( ( 1 ɛ 1 (2 + 1)ɛ 2n )) =0 whch s exactly (H3) for k + 1 Ths fnshes the constructon of step k + 1 and proves the nducton step Now, suppose that ths coverng process contnues untl step n, and that we have D n 1 > ɛ D n 2 Usng (H3) for k = n, we obtan (44) D n n 1 =0 ( 1 ɛ ( 1 (2 + 1)ɛ 2n )) The next step s to show that for n large enough (only dependng on ɛ) we get D n ɛ From Inequalty (44), we deduce: (45) D n ( 1 ɛ(1 (2n 1)ɛ n+1 ) ) n Snce lm (2 1)ɛ +1 = 0 and lm (1 ɛ 2 ) = 0, there s an nteger n 0 such that for all n 0, we have (2 1)ɛ and (1 ɛ 2 ) ɛ If n n 0, Inequalty (45) mples D n (1 ɛ 2 )n ɛ From now, we suppose that the nteger n fxed at the begnnng of the proof s greater than ths n 0 Let us recall what we proved: for all subset D of G satsfyng α(d, K j ) ɛ 2n for all 1 j n, there s an nteger k 0 (wth 1 k 0 n) such that D k0 ɛ More precsely, the proporton of D covered by the sets of the followng ɛ-dsjont famles (K n g) g Rn, (K n 1 g) g Rn 1,, (K n k0+1g) g Rn k0 +1, s at least 1 ɛ Usng ths cover, we want to obtan a good upper bound for h(d)/ To smplfy the notatons, let J = {n k 0 + 1,, n} and wrte K j R j for g R j K j g

10 10 FABRICE KRIEGER for all j J From now, we also wll use subaddtvty and rght-nvarance of the functon h Snce D = j J K j R j D k0 wth we deduce (46) h(d) h( j J K jr j ) D k0 ɛ, + h(d k 0 ) h( j J K jr j ) We obtan h( j J K jr j ) h(k j g) = j J g R j j J Usng condton (C1), we deduce (47) h(k j ) K j g R j h( j J K jr j ) (λ + ɛ) K j g j J g R j + ɛh(1 G ) K j g Remark that the famly contanng the sets K j g, wth j J and g R j, s an ɛ-dsjont famly of D Accordng to Lemma 31 we get (48) K j g 1 ɛ j J g R j Thus, nequaltes (47) and (48) mply (49) h( j J K jr j ) Now, usng nequaltes (46) and (49) we have (410) h(d) λ + ɛ 1 ɛ λ + ɛ 1 ɛ + ɛh(1 G) Snce (F ) s a Følner net, there exsts 0 I such that ( 0 ) α(f, K j ) ɛ 2n for all 1 j n Note that the lmt superor µ of the bounded net ( h(f ) ) F exsts and s the bggest cluster pont of ths net In partcular, there exsts a subnet ( h(f ϕ(j) )) F ϕ(j) j J convergng to µ Let j 0 J such that ϕ(j) 0 for all j j 0 Usng nequalty (410) wth D = F ϕ(j) and for j j 0, we deduce µ = lm j h(f ϕ(j) ) F ϕ(j) λ + ɛ 1 ɛ + ɛh(1 G) Snce the latter nequalty s satsfed for all ɛ ]0, 1 2 ], we can take the lmt when ɛ tends to 0 and we get lm sup h(f ) F endng the proof of the theorem = µ λ = lm nf h(f ) F,

11 ORNSTEIN-WEISS LEMMA 11 References [CoK] M Coornaert, F Kreger, Mean topologcal dmenson for actons of dscrete amenable groups, Dscrete and Contnuous Dynamcal Systems, Volume 13, Number 3, August 2005, [DuS] N Dunford, JT Schwartz, Lnear operators Part I General theory, Pure and appled mathematcs, Wley Classcs Lbrary, New York, 1988 [Føl] E Følner, On groups wth full Banach mean value, Math Scand 3 (1955), [Gre] F P Greenleaf, Invarant means on topologcal groups and ther applcatons, Van Nostrand, 1969 [Gro] M Gromov, Topologcal nvarants of dynamcal systems and spaces of holomorphc maps, Part I, Math Phys Anal Geom 2 (1999), [Ke] John LKelley, General Topology, Graduate Texts n Mathematcs 27, Sprnger-Verlag, New- York, 1975 [Kr] F Kreger, Le lemme d Ornsten-Wess d après Gromov, Dynamcs, Ergodc Theory, and Geometry (ed Bors Hasselblatt), p99-111, Math Sc Res Inst Publ, 54, Cambrdge Unv Press, Cambrdge, 2007 [LW] E Lndenstrauss, B Wess, Mean topologcal dmenson, Israel J Math 115 (2000), 1 24 [Mou] J Mouln Ollagner, Ergodc Theory and Statstcal Mechancs, Lecture Notes n Mathematcs Vol 1115, Sprnger-Verlag, Berln, 1985 [OlS] A Yu Ol shansk, M V Sapr, Non-amenable fntely presented torson-by-cyclc groups, Publ Math, Inst Hautes Étud Sc 96 (2002), [OrW] D S Ornsten, B Wess, Entropy and somorphsm theorems for actons of amenable groups, J Analyse Math 48 (1987), [Pat] A Paterson, Amenablty, Mathematcal Surveys and Monographs 29, Amercan Mathematcal Socety, Provdence, RI, 1988 Max-Planck-Insttut für Mathematk, Vvatsgasse 7, Bonn E-mal address: kregerfabrce@gmalcom