We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers a question that appeared in an early version of the erratum of Birman and Hilden [2].
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 covering space. Their results provide a bridge between the theories of mapping class groups and braid groups. We survey the work of Birman and Hilden, give an overview of the subsequent developments, and discuss open questions and new directions.
Abstract. We say that a cover of surfaces S → X has the Birman-Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a cover has this property. We give new explicit examples of irregular branched covers that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.
We give a simple geometric algorithm that can be used to determine whether or not a post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree for the polynomial, hence determining the polynomial. If it is not, the algorithm produces a Levy cycle, certifying that the map is not equivalent to a polynomial. Our methods are rooted in geometric group theory: we consider a lifting map on a simplicial complex of isotopy classes of trees. As an application, we give a self-contained solution to Hubbard's twisted rabbit problem, which was originally solved by Bartholdi-Nekrashevych using iterated monodromy groups. We also state and solve a generalization of the twisted rabbit problem to the case where the number of post-critical points is arbitrarily large.
For any surface Σ of infinite topological type, we study the Torelli subgroup I(Σ) of the mapping class group MCG(Σ), whose elements are those mapping classes that act trivially on the homology of Σ. Our first result asserts that I(Σ) is topologically generated by the subgroup of MCG(Σ) consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman [4], Powell [24], and Putman [25] we deduce that I(Σ) is topologically generated by separating twists and bounding pair maps. Next, we prove the abstract commensurator group of I(Σ) coincides with MCG(Σ). This extends the results for finite-type surfaces [9,6,7,16] to the setting of infinite-type surfaces.
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