On groups of diffeomorphisms of the interval with finitely many fixed points II. Azer Akhmedov

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1 O groups of diffeomorphisms of the iterval with fiitely may fixed poits II Azer Akhmedov Abstract: I [6], it is proved that ay subgroup of Diff ω +(I) (the group of orietatio preservig aalytic diffeomorphisms of the iterval) is either metaabelia or does ot satisfy a law. A stroger questio is asked whether or ot the Girth Alterative holds for subgroups of Diff ω +(I). I this paper, we aswer this questio affirmatively for eve a larger class of groups of orietatio preservig diffeomorphisms of the iterval where every o-idetity elemet has fiitely may fixed poits. We show that every such group is either affie (i particular, metaabelia) or has ifiite girth. The proof is based o sharpeig the tools from the earlier work [1]. 1. Itroductio Throughout this paper we will write Φ (resp. Φ diff ) to deote the class of subgroups of Γ Homeo + (I) (resp. Γ Diff + (I)) such that every o-idetity elemet of Γ has fiitely may fixed poits. A importat class of such groups is provided by Diff ω +(I) - the group of orietatio preservig aalytic diffeomorphisms of I. Iterestigly, ot every group i Φ is cojugate (or eve isomorphic) to a subgroup of Diff ω +(I), see [3]. We will cosider a atural metric o Homeo + (I) iduced by the C 0 -metric by lettig f = sup f(x) x. x [0,1] For a iteger N 0 we will write Φ N (resp. Φ diff N ) to deote the class of subgroups of Homeo + (I) (resp. Diff + (I)) where every oidetity elemet has at most N fixed poits. It has bee proved i [1] that, for N 2, ay subgroup of Φ diff N of regularity C1+ɛ is ideed solvable, moreover, i the regularity C 2 we ca claim that it is metaabelia. I [3], we improve these results ad give a complete classificatio of subgroups of Φ diff N, N 2, eve at C1 -regularity. There, it is also show that the preseted classificatio picture fails i the cotiuous category, i.e. withi the larger class Φ N. The mai result of this paper is the followig Theorem 1.1. Let Γ Diff + (I) such that every o-idetity elemet has fiitely may fixed poits. The either Γ is isomorphic to a subgroup of Aff + (R) for some 1, or it has ifiite girth. 1

2 2 Remark 1.2. I particular, we obtai that a subgroup of Diff + (I) where every o-idetity diffeomorphism has fiitely may fixed poits, satisfies o law uless it is isomorphic to a subgroup of Aff + (R) for some 1. This also implies a positive aswer to Questio (v) from [6]. If Γ is irreducible (i.e. it does ot have a global fixed poit i (0, 1)), the oe ca take = 1. The proof of Theorem 1.1 relies o the study of diffeomorphism groups which act locally trasitively. To be precise, we eed the followig defiitios. Defiitio 1.3. A subgroup Γ Diff + (I) is called locally trasitive if for all p (0, 1) ad ɛ > 0, there exists γ Γ such that γ < ɛ ad γ(p) p. Defiitio 1.4. A subgroup Γ Diff + (I) is called dyamically 1- trasitive if for all p (0, 1) ad for every o-empty ope iterval J (0, 1) there exists γ Γ such that γ(p) J. The otio of dyamical k-trasitivity is itroduced i [2] where we also make a simple observatio that local trasitivity implies dyamical 1-trasitivity. Notice that if the group is dyamically k-trasitive for a arbitrary k 1 the it is dese i C 0 -metric. Dyamical k- trasitivity for some high values of k (eve for k 2) is usually very hard, if possible, to achieve, ad would be immesely useful (all the results obtaied i [2] are based o just establishig the dyamical 1- trasitivity). I this paper, we would like to itroduce a otio of weak trasitivity which turs out to be sufficiet for our purposes but it is also iterestig idepedetly. Defiitio 1.5. A subgroup Γ Diff + (I) is called weakly k-trasitive if for all g Γ, for all k poits p 1,..., p k (0, 1) with p 1 < < p k, ad for all ɛ > 0, there exist γ Γ such that g(γ(p i )) (γ(p i 1 ), γ(p i+1 )), for all i {1,..., k} ad γ(p k ) < ɛ where we assume p 0 = 0, p k+1 = 1. We say Γ is weakly trasitive if it is weakly k-trasitive for all k 1. The proof of the mai theorem will follow immediately from the followig four propositios which seem iterestig to us idepedetly. Propositio 1.6. Ay irreducible subgroup of Φ diff is either isomorphic to a subgroup of Aff + (R) or it is locally trasitive. Propositio 1.7. Ay irreducible subgroup of Φ diff is either isomorphic to a subgroup of Aff + (R) or it is weakly trasitive.

3 Propositio 1.8. Ay locally trasitive irreducible subgroup of Φ diff is either isomorphic to a subgroup of Aff + (R) or it has ifiite girth. Propositio 1.9. For ay N 1, ay irreducible subgroup of Diff + (I) where every o-idetity elemet has at most N fixed poits is isomorphic to a subgroup of Aff + (R). Let us poit out that, i C 1+ɛ -regularity, Propositio 1.9 is proved i [2] uder somewhat weaker coclusio, amely, by replacig the coditio isomorphic to a subgroup of Aff + (R) with solvable, ad i C 2 -regularity by replacig it with metabelia. Subsequetly, aother (simpler) argumet for this result is give i the aalytic category, by A.Navas [6]. I full geerality ad stregth, Propositio 1.9 is proved i [3]. Thus we eed to prove oly Propositio 1.6, 1.7 ad 1.8. The first of these propositios is proved i Sectio 2, by modifyig the mai argumet from [1]. The secod ad third propositios are proved i Sectio 3; i the proof of the third propositio, we follow the mai idea from [4]. Notatios: For all f Homeo + (I) we will write F ix(f) = {x (0, 1) f(x) = x}. The group Aff + (R) will deote the group of all orietatio preservig affie homeomorphisms of R, i.e. the maps of the form f(x) = ax + b where a > 0. The cojugatio of this group by a orietatio preservig homeomorphism φ : I R to the group of homeomorphisms of the iterval I will be deoted by Aff + (I) (we will drop the cojugatig map φ from the otatio). By choosig φ appropriately, oe ca also cojugate Aff + (R) to a group of diffeomorhisms of the iterval as well, ad (by fixig φ) we will refer to it as the group of affie diffeomorphisms of the iterval. If Γ is a subgroup from the class Φ the oe ca itroduce the followig biorder i Γ: for f, g Γ, we let g < f if g(x) < f(x) ear zero. If f is a positive elemet the we will also write g << f if g < f for every iteger ; we will say that g is ifiitesimal w.r.t. f. For f Γ, we write Γ f = {γ Γ : γ << f} (so Γ f cosists of diffeomorphisms which are ifiitesimal w.r.t. f). Notice that if Γ is fiitely geerated with a fixed fiite symmetric geeratig set, ad f is the biggest geerator, the Γ f is a ormal subgroup of Γ, moreover, Γ/Γ f is Archimedea, hece Abelia, ad thus we also see that [Γ, Γ] Γ f. 3

4 4 2. C 0 -discrete subgroups of Diff + (I): stregtheig the results of [1] Let us first quote the followig theorem from [1]. Theorem 2.1 (Theorem A). Let Γ Diff + (I) be a subgroup such that [Γ, Γ] cotais a free semigroup o two geerators. The Γ is ot C 0 -discrete, moreover, there exists o-idetity elemets i [Γ, Γ] arbitrarily close to the idetity i C 0 metric. I [2], Theorem A is used to obtai local trasitivity results i C 2 - regularity (C 1+ɛ -regularity ) for subgroups from Φ diff N which have derived legth at least three (at least k(ɛ)). However, we eed to obtai local trasitivity results for subgroups which are 1) o-abelia (ot ecessarily o-metaabelia); 2) from a larger class Φ diff ; 3) ad have oly C 1 -regularity. To do this, first we observe that withi the class Φ diff, the coditio Γ cotais a free semigroup by itself implies that [Γ, Γ] is either Abelia or cotais a free semigroup. Propositio 2.2. Let Γ be a o-abelia subgroup from the class Φ diff. The either Γ is locally trasitive or [Γ, Γ] cotais a free semigroup o two geerators. For the proof of the propositio, we eed the followig Defiitio 2.3. Let f, g Homeo + (I). We say the pair (f, g) is crossed if there exists a o-empty ope iterval (a, b) (0, 1) such that oe of the homeomorphisms fixes a ad b but o other poit i (a, b) while the other homeomorphism maps either a or b ito (a, b). It is a well kow folklore result that if (f, g) is a crossed pair the the subgroup geerated by f ad g cotais a free semigroup o two geerators (see [7]). Proof of Propositio 2.2. Without loss of geerality we may assume that Γ is irreducible. Let us first assume that Γ is metaabelia, ad let N be a otrivial Abelia ormal subgroup of Γ. By Hölder s Theorem, there exists f Γ such that F ix(f). O the other had, by irreducibility of Γ, F ix(g) = for all g N\{1}. By Lemma 6.2 i [5], N is ot discrete. Hece, for all ɛ > 0, there exists ω N such that ω < ɛ. This implies that Γ is locally trasitive.

5 Let us ow assume that Γ has derived legth more tha 2 (possibly ifiity). The Γ (1) = [Γ, Γ] is ot Abelia. The by Hölder s Theorem there exists two elemets f, g Γ (1) such that F ix(f) F ix(g). We may assume that (by switchig f ad g if ecessary) there exists p, q F ix(f) {0, 1} such that F ix(f) (p, q) =, F ix(g) (p, q). Without loss of geerality we may also assume that f(x) > x, x (p, q) ad g(p) p. If g(p) > p the f ad g form a crossed pair. But if g(p) = p the for sufficietly big, f ad g form a crossed pair. To fiish the proof of Propositio 1.6 it remais to show the followig Theorem 2.4. Let Γ Φ diff be a subgroup cotaiig a free semigroup i two geerators, such that f (0) = f (1) = 1 for all f Γ. The Γ is locally trasitive. Proof. The proof is very similar to the proof of Theorem A from [1] with a crucial extra detail. Without loss of geerality, we may assume that Γ is irreducible. Let p (0, 1), ɛ > 0, M = ( f (x) + g (x) ). Let also N N ad sup 0 x 1 δ > 0 such that 1/N < mi{ɛ, p, 1 p} ad for all x [1 δ, 1], the iequality φ (x) 1 < 1/10 holds where φ {f, g, f 1, g 1 }. Let W = W (f, g) be a elemet of Γ such that ({f i W (1/N) 2 i 2} {g i W (1/N) 2 i 2}) [1 δ, 1] ad let m be the legth of the reduced word W. Let also x i = i/n, 0 i N ad z = W (1/N). By replacig the pair (f, g) with (f 1, g 1 ) if ecessary, we may assume that f(z) z. The we ca defie (see the proof of Theorem A i [1] for the details) α, β Γ ad a poit z 0 (1 δ, 1) such that the followig coditios hold: (i) α ad β are positive words i f, g of legth at most two, (ii) z 0 = z or z 0 = fg(z), (iii) z 0 z, (iv) z 0 α(z 0 ) βα(z 0 ). The sup ( α (x) + β (x) ) M 2, ad the legth of the word W 0 x 1 i the alphabet {α, β, α 1, β 1 } is at most 2m. Now, for every N, let S = {U(α, β)βα U(α, β) is a positive word i α, β of legth.} 5

6 6 S = 2. Applyig Lemma 1 from [1] to the pair {α, β} we obtai that V (z 0 ) z 0 for all V S. Now, let c k = (k+1) for all k 1. The there exists a sufficietly big such that the followig two coditios are satisfied: (i) there exist a subset S (1) S such that S (1) > (2 c 1 ), ad for all g 1, g 2 S (1), g 1 W (x) g 2 W (x) < 1 (1.9), x {x i 1 i N 1} {p}. (ii) M 2m+4 (1.1) 1 (1.9) < ɛ. The from Mea Value Theorem we obtai that (g 1 W ) 1 (g 2 W )(x) x < 2ɛ for all x [0, 1]. 1 If g 1 W (p) g 2 W (p) for some g 1, g 2 S (1) the we are doe. Otherwise we defie the ext set S (2) {U(α, β)gw g S (1), U(α, β) is a positive word i α, β of legth.} such that S (2) > (2 c 2 ), ad for all g 1, g 2 S (2), g 1 W (x) g 2 W (x) < 1 (1.9), x {x i 1 i N 1} {p}. Agai, by Mea Value Theorem, we obtai that (g 1 W ) 1 (g 2 W )(x) x < 2ɛ 1 to explai this, we borrow the followig computatio from [1]: let h 1 = g 1 W, h 2 = g 2 W, y i = W (x i ), z i = g 1 (y i ), z i = g 2 (y i ), 1 i N. The for all i {1,..., N 1}, we have h 1 1 h 2(x i ) x i = (g 1 W ) 1 (g 2 W )(x i ) x i = (g 1 W ) 1 (g 2 W )(x i ) (g 1 W ) 1 (g 1 W )(x i ) = W 1 g 1 1 g 2(y i ) W 1 g 1 1 g 1(y i ) = W 1 g 1 1 (z i ) W 1 g 1 1 (z i) Sice g 1, g 2 S, by the Mea Value Theorem, we have h 1 1 h 2(x i ) x i M 2m+4 (1.1) z 1 z 1 < M 2m+4 (1.1) 1 (1.9) Sice m is fixed, for sufficietly big we obtai that h 1 1 h 2(x i ) x i < ɛ. The h 1 1 h 2(x) x < 2ɛ for all x [0, 1].

7 for all g 1, g 2 S (2) g 1, g 2 S (2), x [0, 1]. Agai, if g 1 W (p) g 2 W (p) for some the we are doe, otherwise we cotiue the process as S (k) are chose ad g 1 W (p), the we choose follows: if the sets S S (1) g 2 W (p) for all g 1, g 2 S (k) S (k+1) {U(α, β)g g S (k), U(α, β) is a positive word i α, β of legth.} such that S (k+1) > (2 c k+1 ), ad for all g 1, g 2 S (k+1), g 1 W (x) g 2 W (x) < 1 (1.9), x {x i 1 i N 1} {p}. Sice Γ belogs to the class Φ diff the process will stop after fiitely may steps, ad we will obtai a elemet ω with orm less tha 2ɛ such that ω(p) p Weak Trasitivity I this sectio we prove Propositio 1.7 ad the Propositio 1.8. First, we eed the followig lemma which follows immedaitely from the defiitio of Γ f. Lemma 3.1. Let Γ be a fiitely geerated irreducible subgroup of the class Φ diff, f be the biggest geerator of Γ with at least oe fixed poit, z = mi F ix(f), ad ω(z) > z for some ω Γ f. The there exists ɛ > 0 such that if 0 < a < b < ɛ, f p (a) < b for some p 4, the, for sufficietly big, f p 4 (ωf ω 1 (a)) < ωf ω 1 (b). Proof of Propositio 1.7. Let Γ be a irreducible o-affie subgroup of Φ diff, g Γ, ad 0 < p 1 < < p k < 1. We may assume that Γ is fiitely geerated. The, let f be the biggest geerator of Γ. Without loss of geerality we may also assume that f has at least oe fixed poit ad mi F ix(f) < p 1. By dyamical 1-trasitivity (i.e. Propositio 1.6), there exist elemets ω 1,..., ω k Γ f such that mi F ix(η i ) (p i, p i+1 ), 1 i k where η i = ω i... ω 1 fω ω 1 i. We ca choose q 4k such that f q (x) > g(x) for all x (0, 1 mi F ix(f)). 2 Now, applyig Lemma 3.1 iductively, for sufficietly small ɛ > 0 ad sufficietly big, we obtai that f 4q 4 (η 1 (x)) < η 1 (y), for all x, y (0, ɛ) where f 4q (x) < y

8 8 f 4q 8 (η 2 (p 1 ) < η 2 (p 2 ),... f 4q 4i (η i (p j ) < η i (p j+1 ), 1 j i 1 f 4q 4k (η k (p j)) < η k (p j+1), 1 j k 1. Thus it suffices to take γ = η, ad we obtai that k... γ(p j 1 ) < f q (γ(p j )) < γ(p j+1 ), 1 j k 1 where we assume p 0 = 1. The, we have γ(p j 1 ) < g(γ(p j )) < γ(p j+1 ), 1 j k 1. Now we are ready to prove Propositio 1.8. Let Γ be a fiitely geerated irreducible o-affie subgroup of Homeo + (I). Let S = {f 1,..., f s } be a geeratig set of Γ such that S S 1 = (i particular, S does ot cotai the idetity elemet), f be the biggest geerator of S S 1, ad let also m 10s. By the defiitio of the order, there exists ɛ > 0 such that f(x) ξ(x) for all x (0, ɛ), ξ S S 1. By Propositio 1.9 we ca choose g Γ such that F ix(g) > 8m. Sice Γ is irreducible by local trasitivity ad weak trasitivity, we ca fid h Γ ad ɛ > 0 such that the followig coditios hold: (i) γ(spa(hgh 1 )) (0, ɛ) for all γ B 2m (1), (ii) if F ix(hgh 1 ) = {p 1,..., p k } the p i 1 < f 2m (p i ) < p i+1, 2 i k 1. (iii) F ix(f i ) {p 1,..., p k } =, for all i {1,..., s}. Now, let θ = hgh 1 ad Let also S = {θ, θ m f 1 θ m, θ 2m f 2 θ 2m,..., θ sm f s θ sm }. δ = 1 5 mi{p i+1 p i 1 i k 1} ad V = (p i δ, p i + δ). 2 i k 1

9 We ow let x = p j+p j+1 where j = [ k ]. The x / V, ad for a 2 2 sufficietly big, by a stadard pig-pog argumet, for ay otrivial word W i the alphahbet S of legth at most m, we obtai that W (x) V, hece W (x) x. Sice m is arbitrary, we coclude that girth(γ) =. 9 Refereces 1. Akhmedov A. A weak Zassehaus lemma for subgroups of Diff(I). Algebraic ad Geometric Topology. vol.14 (2014) Akhmedov A. Extesio of Hölder s Theorem i Diff 1+ɛ + (I). Ergodic Theory ad Dyamical Systems, to appear Akhmedov, A. O groups of diffeomorphism of the iterval with fiitely may fixed poits I. 4. Akhmedov, A. Girth Alterative for subgroups of P L o (I). Preprit Akhmedov, A. O the height of subgroups of Homeo + (I), Joural of Group Theory, 18, o.1, (2015) Navas, A. Groups, Orders ad Laws. To appear i Groups, Geometry, Dyamics Navas, A. Groups of Circle Diffemorphisms. Chicago Lectures i Mathematics, Azer Akhmedov, Departmet of Mathematics, North Dakota State Uiversity, Fargo, ND, 58102, USA address: azer.akhmedov@dsu.edu