Finiteness results for Teichmueller curves
  We show that for each genus there are only finitely many algebraically primitive Teichmueller curves C, such that i) C lies in the hyperelliptic locus and ii) C is generated by an abelian differential with two zeros of order g-1.   We prove moreover that for these Teichmueller curves the trace field of the affine group is not only totally real but cyclotomic. 
Martin Moeller