Trace fields of representations and virtually Haken 3-manifolds

Author: Zhang, Xingru

Source: Quarterly Journal of Mathematics, Volume 56, Number 3, September 2005 , pp. 431-442(12)

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Abstract:

We introduce and study trace fields and invariant trace fields of SL<inf>2</inf>(C) and PSL<inf>2</inf>(C) representations of 3-manifold groups. We give conditions on such fields which imply that the underlying 3-manifold is virtually Haken or has virtually positive or infin

first Betti number. In particular, we define the notion of an algebraically trace-proper surface subgroup in a 3-manifold group, and show that any closed orientable irreducible 3-manifold with such a surface subgroup is virtually Haken. We give infinitely many families of closed orientable hyperbolic non-Haken 3-manifolds with algebraically trace-proper surface subgroups.

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Document Type: Research article

DOI: http://dx.doi.org/10.1093/qmath/hah041

Publication date: 2005-09-01

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The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. Areas such as algebra, differential geometry, and global analysis receive particular emphasis. However the journal avoids specialization.