2020/10/12 by Sang-hyun Kim, Sang‐Hyun Kim, Kim, Sang-hyun +4
Mathematics · #Abelian group #Action (physics) #Advanced Operator Algebra Research #Combinatorics #Corollary #Cyclic group #Discrete mathematics #Disjoint sets #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group (periodic table) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Interval (graph theory) #Mathematics #Physics #math.DS #math.GR
paper · pdf · doi:10.48550/arxiv.2010.05722
22 pages, to appear in J. Mod. Dyn
openalex publication_date 2020/10/12 · arxiv created 2021/04/07 · arxiv updated 2021/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if H and K are two non-solvable groups then a faithful C1,τ action of H× K on a compact interval I is \em not overlapping for all τ>0, which by definition means that there must be non-trivial h∈ H and k∈ K with disjoint support. As a corollary we prove that the right-angled Artin group (F2× F2)*ℤ has critical regularity one, which is to say that it admits a faithful C1 action on I, but no faithful C1,τ action. This is the first explicit example of a group of exponential growth which is without nonabelian subexponential growth subgroups, whose critical regularity is finite, achieved, and known exactly. Another corollary we get is that Thompson's group F does not admit a faithful C1 overlapping action on I, so that F*ℤ is a new example of a locally indicable group admitting no faithful C1--action on I.