Existence and genericity of finite topological generating sets for homeomorphism groups

Abstract

We show that the topological groups Diff 1 + (I) and Diff 1 + (S 1 ) of orientation-preserving C 1 -diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite topological generating sets in related groups. We show that the generic pair of elements in the homeomorphism group Homeo + (I) generate a dense subgroup of Homeo + (I). By contrast, if M is any compact connected manifold with boundary other than the interval, we observe that an open dense set of pairs from the associated boundary-fixing homeomorphism group Homeo 0 (M, ∂M ) will generate a discrete subgroup. We make similar observations for homeomorphism groups of manifolds without boundary including S 1 . arXiv:1508.04604v2 [math.GR]

Generation and genericity of the group of absolutely continuous homeomorphisms of the interval

“…(We prove this result as Theorem 6.) The proof employs the main techniques of the proof that H + is generically 2-generated, due to Akhmedov & Cohen [AC19], tailored to the absolutely continuous case.…”

“…As mentioned in the introduction, the main technique is similar to the proof that g-rk (H + ) = 2, from [AC19]. The main difference in the H AC + case is the extra care that must be taken to ensure various approximations by "nice" maps can be made.…”