Relative Weight Filtrations on Completions of Mapping Class Groups

Abstract

This paper gives an exposition of relative weight filtrations on completions of mapping class groups associated to a stable degeneration of marked genus g curves. These relative weight filtrations have been constructed using Galois theory (with Matsumoto) and Hodge theory (with Pearlstein and Terasoma). It is shown that the level 0 part of the relative weight filtration is an analogue of a parabolic subalgebra of a Kac-Moody Lie algebra. It is shown that all such subalgebras correspond to equivalence classes of pants decompositions of the surface -- two being equivalent if and only if they determine the same handlebody that the reference surface bounds. One application is to show that handlebody subgroups of mapping class groups contain elements arbitrarily far down the lower central series of Torelli groups. (This result was also obtained independently by Jamie Jorgensen.)

References in corpus (2)

• Cohomological Aspects of Magnus Expansions
• Geometric Proofs of Some Results of Morita

Cited by in corpus (8)

• Notes on the Universal Elliptic KZB Equation
• The handlebody group and the images of the second Johnson homomorphism
• Relative Pro-$\ell$ Completions of Mapping Class Groups
• Cohomology Jumping Loci and Relative Malcev Completion
• Rational points of universal curves in positive characteristics
• Quasi-homomorphisms on mapping class groups vanishing on a handlebody group
• On the sections of universal hyperelliptic curves
• On the completion of the mapping class group of genus two