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Cohomological aspects of magnus expansions.
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In this article, the authors generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group Aut(Fn).Abstract:
We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group Aut(Fn).The extended ones are not homomor- phisms, but satisfy an infinite sequence of coboundary relations, so that we call them the Johnson maps.In this paper we confine ourselves to studying the first and the second relations, which have cohomological consequences about the group Aut(Fn) and the mapping class groups for surfaces.The first one means that the first Johnson map is a twisted 1-cocycle of the group Aut(Fn).Its cohomology class coincides with "the unique elementary particle" of all the Morita-Mumford classes on the mapping class group for a surface (Ka1) (KM1).The second one restricted to the mapping class group is equal to a fundamental relation among twisted Morita-Mumford classes pro- posed by Garoufalidis and Nakamura (GN) and established by Morita and the author (KM2).This means we give a coherent proof of the fundamental relation.The first Johnson map gives the abelianization of the induced automorphism group IAn of a free group in an explicit way.read more
Citations
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FI-modules and stability for representations of symmetric groups
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FI-modules and stability for representations of symmetric groups
TL;DR: The theory of FI-modules is introduced and developed in this paper, where the authors show that for any fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n.
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Basis-conjugating automorphisms of a free group and associated Lie algebras
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References
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Towards an Enumerative Geometry of the Moduli Space of Curves
TL;DR: In this paper, a Chow ring for the moduli space M g of curves of genus g and its compactification M g is defined, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci in M g in terns of these classes.
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Stable real cohomology of arithmetic groups
TL;DR: In this article, it was shown that if a certain quadratic form depending on q is positive non-degenerate, then any Γ-invariant harmonic q-form is automatically G-Invariant.
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Moduli of graphs and automorphisms of free groups
Marc Culler,Karen Vogtmann +1 more
TL;DR: In this article, the authors study the outer-to-morphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study mapping classes of surfaces, by studying the act ion on a space X, which is analogous to the Teichmtiller space of hyperbol ic metrics on a surface; the points of X, are metric structures on graphs with fundamental group F. The 0cells are called nodes and the l-cells edges.