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Goldman Bracket : Center, Geometric Intersection Number & Length Equivalent Curves

Electronic Theses of Indian Institute of Science

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Title Goldman Bracket : Center, Geometric Intersection Number & Length Equivalent Curves
 
Creator Kabiraj, Arpan
 
Subject Lie Algebra Structure
Vector Spaces
Curves on Oriented Spaces
Length Equivalent Curves
Goldman Lie Algebra
Goldman Bracket
Hochschild Cohomology
Hyperbolic Geometry
Mathematics
 
Description Goldman [Gol86] introduced a Lie algebra structure on the free vector space generated by the free homotopy classes of oriented closed curves in any orientable surface F . This Lie bracket is known as the Goldman bracket and the Lie algebra is known as the Goldman Lie algebra. In this dissertation, we compute the center of the Goldman Lie algebra for any hyperbolic surface of finite type. We use hyperbolic geometry and geometric group theory to prove our theorems. We show that for any hyperbolic surface of finite type, the center of the Goldman Lie algebra is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures.

We use these results to identify the quotient of the Goldman Lie algebra of a non-closed surface by its center as a sub-algebra of the first Hochschild cohomology of the fundamental group.
Using hyperbolic geometry, we prove a special case of a theorem of Chas [Cha10], namely, the geometric intersection number between two simple closed geodesics is the same as the number of terms (counted with multiplicity) in the Goldman bracket between them.
We also construct infinitely many pairs of length equivalent curves in any hyperbolic surface F of finite type. Our construction shows that given a self- intersecting geodesic x of F and any self-intersection point P of x, we get a sequence of such pairs.
 
Contributor Gadgil, Siddhartha
 
Date 2017-11-30T05:02:03Z
2017-11-30T05:02:03Z
2017-11-30
2016
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/2838
http://etd.ncsi.iisc.ernet.in/abstracts/3689/G27288-Abs.pdf
 
Language en_US
 
Relation G27288