Fundamental Groups and Bounds for the Extremal Length
摘要
In this chapter we consider any element of the fundamental group of the twice punctured complex plane and give effective upper and lower bounds for its extremal length with totally real horizontal boundary values. The upper and lower bounds differ by a multiplicative constant not depending on the element. The estimates are given in terms of a natural syllable decomposition of the reduced word representing the element. In the last section (Sect. 9.7) we give similar estimates for pure 3-braids. The extremal length with totally real horizontal boundary values is capable to give more subtle information regarding Gromov’s Oka Principle and limitations of its validity than the respective invariant of conjugacy classes. Moreover, for any pure 3-braid the provided estimates of its extremal length with totally real horizontal boundary values imply estimates of the extremal length of its conjugacy class, and hence, of its entropy.