Questions and remarks on discrete and dense subgroups of Diff(I)

Abstract

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.

is not a Polish group

“…The first group is of a regularity class less than C 1 , while the second lies strictly between C 1 -and C 2 -regularity. The dynamical properties of these groups and their countable subgroups have received considerable attention in the literature-for some overview of facts and open problems, we refer the reader to the well-known reference [13] or the recent survey article [1]. Our results indicate that these groups are fundamentally unlike the C kgroups; they prohibit Polish topologization entirely.…”

“…Suppose for a contradiction that G carries some Polish group topology. In cases (1) and 2 For an element g ∈ G, let C(g) denote the centralizer of g. Each C(g) is closed in G, as centralizers are closed in any Hausdorff topological group. The set X = {g ∈ PL + (R) : g fixes [0, 1] pointwise}.…”

Finiteness of the topological rank of diffeomorphism groups

“…Proof. First, let us notice that, since u is a C k+1 -flat polynomial, for some K 0 > 0 independent of N, |u (1)…”

“…We are also motivated to understand discrete and dense finitely generated subgroups of these groups, seeking parallels with the very rich and mostly established theory of discrete and dense subgroups of Lie groups. The reader may consult the paper [1] for a large number of remarks and open questions in this program where mainly the group Diff + (I) has been considered, however, quite many of the questions there are still meaningful for a general compact smooth manifold and in the higher regularity as well.…”

Existence and genericity of finite topological generating sets for homeomorphism groups

“…This question is especially intriguing since the topological group Diff 1 + (I) is homeomorphic to an infinite dimensional separable Banach space. 1 The first main result of our paper is to show that, despite a lack of simple examples, dense finitely generated subgroups of Diff 1 + (I) do exist.…”