Braided surfaces and their characteristic maps (Q6108707)

Let \(\Sigma\) be a surface. A braided surface over \(\Sigma\) is a locally flat PL embedding \(j: S \to \Sigma \times \mathbb{R}^2\) of a surface \(S\) such that the composition \(p \circ j\) is a branched covering, where \(p: \Sigma \times \mathbb{R}^2 \to \Sigma\) is the projection. The composition \(p \circ j\) is called the characteristic map. Two braided surfaces \(j_0, j_1: S \to \Sigma\times \mathbb{R}^2\) are equivalent if there exists an ambient isotopy \(h_t: \Sigma \times \mathbb{R}^2 \to \Sigma \times \mathbb{R}^2\) such that \(h_0=id\) and \(h_1 \circ j_0=j_1\) and \(h_t\) is fiber-preserving, that is, there exists a homeomorphism \(\phi_t : \Sigma \to \Sigma\) such that \(p \circ h_t=\phi_t \circ p\); and for two braided surfaces \(j_0, j_1: S \to \Sigma\times \mathbb{R}^2\) with the same finite set \(B\) of the branch loci of the branched coverings \(p \circ j_0\), \(p \circ j_1\), they are strongly equivalent if they are equivalent and moreover \(\phi_t\) is an isotopy fixing pointwise the branch locus \(B\). \newline The main results are as follows. There exists some \(h_{n,m}\) such that any degree \(n\) ramified covering of the closed orientable surface of genus \(g \geq h_{n,m}\) with \(m\) branch points occurs as the composition \(p \circ j\), where \(j: S \to \Sigma \times \mathbb{R}^2\) and \(p: \Sigma \times \mathbb{R}^2 \to \Sigma\) are maps used in the definition of a braided surface. For any \(g \geq 2\), there exist infinitely many simple non-abelian groups \(G\) and surjective homomorphisms \(\pi_1(\Sigma_g) \to G\), where \(\Sigma_g\) is a closed orientable surface of genus \(g\), such that the kernels are invariant by all automorphisms of \(G\) and do not contain any simple loop homotopy class. Further, the authors treat finite dimensional Hermitian representations of braid groups and construct spherical functions on representation spaces associated to discrete groups by pullback of spherical functions defined on Lie groups, and show that strong equivalence classes of braided surfaces are separated by some spherical function if and only if they are profinitely separated. \newline In the paper, the authors discuss braided surfaces and the condition for a homomorphism \(f: \pi_1(\Sigma \backslash B) \to B_n\) to be the braid monodromy, where \(B_n\) is the braid group, and the correspondence to the orbits of the mapping class group action of a punctured surface on the set of homomorphisms \(\pi_1(\Sigma \backslash B) \to B_n\) satisfying a certain condition. And the authors discuss lifting and stable equivalence of surjective homomorphisms of surface groups. Further, they investigate algebraically the thickness \(t(f)\) of the surjective homomorphism \(f: \pi_1(\Sigma_g) \to G\) with \(sc(f)=0\), where \(sc(f)\) is a homological invariant called the Schur class of \(f\), and \(t(f)\) is defined as the smallest value of \(n\) for which there exists a 3-manifold \(M\) with boundary \(\Sigma_g\) and Heegaard genus \(g+n\) such that \(f\) extends to \(F: \pi_1(M) \to G\).