We prove that if Γ is subgroup of Diff 1+ + (I) and N is a natural number such that every non-identity element of Γ has at most N fixed points then Γ is solvable. If in addition Γ is a subgroup of Diff 2 + (I) then we can claim that Γ is metaabelian.It is a classical result (essentially due to Hölder, cf.[N1]) that if Γ is a subgroup of Homeo + (R) such that every nontrivial element acts freely then Γ is Abelian. A natural question to ask is what if every nontrivial element has at most N fixed points where N is a fixed natural number. In the case of N = 1, we do have a complete answer to this question: it has been proved by Solodov (not published), Barbot [B], and Kovacevic [K] that in this case the group is metaabelian, in fact, it is isomorphic to a subgroup of the affine group Aff(R). (see [FF] for the history of this result, where yet another nice proof is presented).In this paper, we answer this question for an arbitrary N assuming some regularity on the action of the group.Our main result is the following theorem.
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.
We prove that if a non-cyclic finitely generated group is hyperbolic (or one-relator, or linear), then it has infinite girth iff it is not virtually solvable.
In recent decades, many remarkable papers have appeared which are devoted to the study of finitely generated subgroups of Diff+([0, 1]) (see [8, 15, 16, 19–23, 29, 30, 39, 40] only for some of the most recent developments). In contrast, discrete subgroups of the group Diff+([0, 1]) are much less studied. Very little is known in this area especially in comparison with the very rich theory of discrete subgroups of Lie groups which has started in the works of F. Klein and H. Poincaré in the 19th century, and has experienced enormous growth in the works of A. Selberg, A. Borel, G. Mostow, G. Margulis and many others in the 20th century. Many questions which are either very easy or have been studied a long time ago for (discrete) subgroups of Lie groups remain open in the context of the infinite-dimensional group Diff+([0, 1]) and its relatives.
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