The Birman-Hilden theory

Abstract

In the 1970s Joan Birman and Hugh Hilden wrote several papers on the problem of relating the mapping class group of a surface to that of a cover. We survey their work, give an overview of the subsequent developments, and discuss open questions and new directions.

arXiv commentFor the journal Celebratio Mathematica (http://celebratio.org)

References in corpus (16)

• The mapping class group of a genus two surface is linear
• Bounded cohomology and non-uniform perfection of mapping class groups
• Injections of mapping class groups
• Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1
• Normal all pseudo-Anosov subgroups of mapping class groups
• On the linearity of certain mapping class groups
• The asymptotic dimension of a curve graph is finite
• Arithmetic Veech sublattices of $\SL(2,\Z)$
• Lifting Homeomorphisms and Cyclic Branched Covers of Spheres
• Image of the Burau Representation at $d$-th Roots of unity
• Noncomplex smooth 4-manifolds with Lefschetz fibrations
• Palindromic Automorphisms of Free Groups
• Mapping class groups of covers with boundary and braid group embeddings
• Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration
• Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue
• On cluster C*-algebras

Cited by in corpus (10)

• Presentations of Dehn quandles
• Commensurations of $\mathrm{Aut}(F_N)$ and its Torelli subgroup
• Curves intersecting in a circuit pattern
• Generating the liftable mapping class groups of regular cyclic covers
• Configuration space, moduli space and 3-fold covering space
• Lifting homeomorphisms and finite abelian branched covers of the 2-sphere
• Endomorphisms of Artin groups of type ${\tilde{A}}_n$
• Generating the liftable mapping class groups of cyclic covers of spheres
• Prym Representations of the Handlebody Group
• Liftable automorphisms of right-angled Artin groups