Goeritz Groups of Bridge Decompositions

Abstract

For a bridge decomposition of a link in the $3$-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the $3$-sphere that preserve each of the bridge sphere and link setwise. After describing basic properties of this group, we discuss the asymptotic behavior of the minimal pseudo-Anosov entropies. We then give an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings of the $3$-sphere and the real projective space.

Distance and the Goeritz groups of bridge decompositions

“…In that paper, a variation of Namazi and Johnson's result for the case of bridge decomposition was obtained: it was shown that the constant C for the finiteness of the Goeritz group can be taken uniformly to be at most 3796. The main result of the paper is the following, which improves the above mentioned result of [6].…”

“…Johnson [9] had refined this result by showing the constant C can be taken to be at most 4 independently of the genus of the Heegaard surface. The concept of Goeritz group has also been extended for bridge decompositions in [6]. In that paper, a variation of Namazi and Johnson's result for the case of bridge decomposition was obtained: it was shown that the constant C for the finiteness of the Goeritz group can be taken uniformly to be at most 3796.…”