Existence and genericity of finite topological generating sets for homeomorphism groups

2015/08/19 by Azer Akhmedov, Akhmedov, Azer, Michael P. Cohen +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #math.DS #math.GR #math.GT

paper · pdf · doi:10.48550/arxiv.1508.04604

Several sentences rephrased for better clarity

openalex publication_date 2015/08/19 · arxiv created 2015/10/14 · arxiv updated 2015/10/15 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We show that the topological groups Diff₊¹(I) and Diff₊¹(\mathbbS¹) of orientation-preserving C¹-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(M,∂ M) will generate a discrete subgroup. We make similar observations for homeomorphism groups of manifolds without boundary including \mathbbS¹.

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