On the simplicity of homeomorphism groups of a tilable lamination (Q328582)

Let \(\Omega\) be a minimal \(\mathbb R^d\)-tilable lamination, i.e., a flat lamination in which each leaf is isometric to \(\mathbb R^d\) and dense in \(\Omega\) and there is a transversal which is a Cantor set. Let \(G\) be either \(\mathcal H_\mathcal L(\Omega)\), the group of leaf-preserving homeomorphisms of \(\Omega\), or \(\mathcal H_{vsp}(\Omega)\), the sub-group of homeomorphisms that also preserve the vertical structure, each with the \(C^0\)-topology. Then the component \(G^0\) of the identity of \(G\) is open in \(G\), is perfect and simple, and is the path component of the identity of \(G\).