The dual boundary complex of the SL2SL_2 character variety of a punctured sphere


Suppose C1,,CkC_1,\ldots , C_k are generic conjugacy classes in SL2(C)SL_2({\mathbb C}). Consider the character variety of local systems on P1{y1,,yk}{\mathbb P}^1-\{ y_1,\ldots , y_k\} whose monodromy transformations around the punctures yiy_i are in the respective conjugacy classes CiC_i. We show that the dual boundary complex of this character variety is homotopy equivalent to a sphere of dimension 2(k3)12(k-3)-1

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