Subgroups of the Torelli group

Let Mod(Sg) be the mapping class group of an orientable surface of genus g, Sg. The action of Mod(Sg) on the homology of Sg induces the well-known symplectic representation: <BR> <BR> <center>Mod(Sg) ---> Sp(2g, Z).</center> <BR> The kernel of this representation is called the Torelli group, I(Sg). <BR> <BR> We will study two subgroups of I(Sg). First we will look at the subgroup generated by all SIP-maps, SIP(Sg). We will show SIP(Sg) is not I(Sg) and is in fact an infinite index subgroup of I(Sg). We will also classify which SIP-maps are in the kernel of the Johnson homomorphism and Birman-Craggs-Johnson homomorphism. <BR> <BR> Then we will look at the symmetric Torelli group, SI(Sg). More specifically, we will investigate the group generated by Dehn twists about symmetric separating curves denoted H(Sg). We will show the well-known Birman-Craggs-Johnson homomorphism is not able to distinguish among SI(Sg), H(Sg), or K(Sg), where K(Sg) is the subgroup generated by Dehn twists about separating curves. Elements of H(Sg) act naturally on the symmetric separating curve complex, CH(S). We will show that when g > 4 <BR> <BR> <center> Aut(CH(Sg)) = SMod^{+/-}(Sg) / < i > </center> <BR> where SMod(Sg) is the symmetric mapping class group and i is a fixed hyperelliptic involution. Lastly we will give an algebraic characterization of Dehn twists about symmetric separating curves.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Applied Mathematics

Grantor

Mathematics

Year dc:date.available

2010

Author dc:creator

• Childers, Leah R

• symmetric separating curve complex
• Torelli group
• mapping class group
• symmetric Torelli group
• simply intersecting pair maps
• curve complex
• symmetric mapping class group

Statement dc:rights

• unrestricted
• Release the entire work immediately for access worldwide.

Identifier

etd-05252010-101928
https://repository.lsu.edu/gradschool_dissertations/536

OAI identifier oai:identifier

oai:repository.lsu.edu:gradschool_dissertations-1535