An important theorem of Ling states that if G is any factorizable non-fixing group of homeomorphisms of a paracompact space then its commutator subgroup [G, G] is perfect. This paper is devoted to further studies on the algebraic structure (e.g. uniform perfectness, uniform simplicity) of [G, G] and [G,G], whereG is the universal covering group of G. In particular, we prove that if G is bounded factorizable non-fixing group of homeomorphisms then [G, G] is uniformly perfect (Corollary 3.4). The case of open manifolds is also investigated. Examples of homeomorphism groups illustrating the results are given. Definition 1.1. Let U be an open cover of X. A group of homeomorphisms G of a space X is called U-factorizable if for every g ∈ G there are g 1 , . . . , g r ∈ G with g = g 1 . . . g r and such that supp(g i ) ⊂ U i , i = 1, . . . , r, for some U 1 , . . . , U r ∈ U. G is called factorizable if for every open cover U of X it is U-factorizable.

Abstract. Given a principal G-bundle π : M → B, let HG(M ) be the identity component of the group of G-equivariant homeomorphisms on M . The problem of the uniform perfectness and boundedness of HG(M ) is studied. It occurs that these properties depend on the structure of H(B), the identity component of the group of homeomorphisms of B, and of B itself. Most of the obtained results still hold in the C r category.

Let M be a manifold, p ∈ M and let H(M, p) be the identity component of the group of all compactly supported homeomorphisms of M fixing p. It is shown that H(M, p) is a perfect group. Next, we prove that the group H(R n , 0) is bounded. In contrast, in the C ∞ category the diffeomorphism group D ∞ (R n , 0), analogous to H(R n , 0), is neither perfect nor bounded. Finally, the boundedness and uniform perfectness of H(M, p) is studied.

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Abstract.We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of a foliated manifold with leafwise Randers structure is considered. In this case the estimates for one dimensional foliation defined by a vector field in terms of the topological entropy of a flow are presented.