A weak Zassenhaus Lemma for discrete subgroups of Diff(I)

Abstract

We prove a weaker version of the Zassenhaus Lemma for subgroups of Diff(I). We also show that a group with commutator subgroup containing a non-abelian free subsemigroup does not admit a C 0 -discrete faithful representation in Diff(I).In this paper, we continue our study of discrete subgroups of Diff + (I) -the group of orientation preserving diffeomorphisms of the closed interval I = [0, 1]. Following recent trends, we try to view the group Diff + (I) as an analog of a Lie group, and we study still basic questions about discrete subgroups of it. This paper can be viewed as a continuation of [A] although the proofs of the results of this paper are independent of [A].Throughout the paper, the letter G will denote the group Diff + (I).On G, we assume the metric induced by the standard norm of the Banach space C 1 [0, 1]. We will denote this metric by d 1 . Sometimes, we also will consider the metric on G that comes from the standard sup norm ||f || 0 = sup x∈[0,1] |f (x)| of C[0, 1] which we will denote by d 0 . However, unless specified, the metric in all the groups Diff r + (I), r ∈ R, r ≥ 1 will be assumed to be d 1 .The central theme of the paper is the Zassenhaus Lemma. This lemma states that in a connected Lie group H there exists an open nonempty neighborhood U of the identity such that any discrete subgroup generated by elements from U is nilpotent (see [R]). For example, if H is a simple Lie group (such as SL 2 (R)), and Γ ≤ H is a lattice, then Γ cannot be generated by elements too close to the identity.In this paper we prove weak versions of the Zassenhaus Lemma for the group G = Diff + (I). Our study leads us to showing that finitely generated groups with exponential growth which satisfy a very mild condition do not admit faithful C 0 -discrete representation in G:Theorem A. Let Γ be a subgroup of G, and f, g ∈ [Γ, Γ] such that f and g generate a non-abelian free subsemigroup. Then Γ is not C 0 -discrete. 1 2We also study the Zassenhaus Lemma for the relatives of G such as Diff 1+c + (I), c ∈ R, c > 0 -the group of orientation preserving diffeomorphisms of regularity 1 + c. In the case of Diff 1+c + [0, 1], combining Theorem A with the results of -bf [N2], we show that C 0 -discrete subgroups are more rare.Theorem B. Let Γ be a C 0 -discrete subgroup of Diff 1+c + [0, 1]. Then Γ is solvable with solvability degree at most k(c).Theorem B can be strengthened if the regularity is increased further; combining Theorem A with the results of Navas [N2], Plante-Thurston [PT], and Szekeres [S] we obtain the following Theorem C. If Γ is C 0 -discrete subgroup of Diff 2 + [0, 1] then Γ is metaabelian.

On groups of diffeomorphisms of the interval with finitely many fixed points I

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We strengthen the results of [1], consequently, we improve the claims of [2] obtaining the best possible results. Namely, we prove that if a subgroup Γ of Diff + (I) contains a free semigroup on two generators then Γ is not C 0 -discrete.

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“…It has been proved in [1] that, for N ≥ 2, any subgroup of Φ diff N of regularity C 1+ǫ is indeed solvable, moreover, in the regularity C 2 we can claim that it is metaabelian. The argument there fails short in complete characterization of subgroups of Φ diff N , N ≥ 2 even at these increased regularities.…”

On the height of subgroups of Homeo+(I)

“…We prove the tower characterization of solvable subgroups in one direction and show that the other direction fails badly in the continuous category. As a byproduct of our method we obtain interesting new results related to (but not covered by) the results of [2] and [3] (See Remarks 3.3 and 6.3 respectively).…”

“…As a corollary of Lemma 6.2 we obtain that a solvable non-Abelian subgroup of Diff ω + (I) is never C 0 -discrete. On the other hand, it is proved in [3] that a non-solvable (non-metaabelian) subgroup of Diff 1+ǫ + (I) (of Diff 2 + (I)) is never C 0 -discrete. Thus, in the context of analytic diffeomorphisms, Lemma 6.2 gives us a new result not covered by the results of [3].…”

“…in Theorem A, as stated in[3], the claim is that if [Γ, Γ] contains a free semigroup then Γ is not discrete, but notice that, first, the only property of the commutator subgroup used in the proof is the fact that an element of [Γ, Γ] has derivative 1 at the endpoints of the interval [0, 1]; second, the elements constructed in the proof which are arbitrarily close to identity in C 0 -metric are indeed from the subgroup [Γ, Γ]. Thus in[3], we have indeed proved the following claim: if Γ contains a free semigroup on generators f 1 , f 2 and f ′ i (0) = f ′ i (1) = 1, i = 1, 2 then the commutator subgroup [Γ, Γ] is not C 0 -discrete.…”