Title page for ETD etd-05252010-101928


Type of Document Dissertation
Author Childers, Leah R
Author's Email Address leah@math.lsu.edu, leah.childers@gmail.com
URN etd-05252010-101928
Title Subgroups of the Torelli Group
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Brendle, Tara Committee Chair
Dani, Pallavi Committee Co-Chair
Dasbach, Oliver Committee Member
Delzell, Charles Committee Member
Lawson, Jimmie Committee Member
Trahan, Jerry Dean's Representative
Keywords
  • symmetric separating curve complex
  • Torelli group
  • mapping class group
  • symmetric Torelli group
  • simply intersecting pair maps
  • curve complex
  • symmetric mapping class group
Date of Defense 2010-05-13
Availability unrestricted
Abstract
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:


Mod(Sg) ---> Sp(2g, Z).


The kernel of this representation is called the Torelli group, I(Sg).



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.



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



Aut(CH(Sg)) = SMod^{+/-}(Sg) / < i >


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.

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