## Parabolic Higgs bundles and the Deligne-Simpson Problem for loxodromic conjugacy classes in PU(n,1)

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##### Date

2017##### Author

Maschal Jr, Robert Allan

##### Advisor

Wentworth, Richard

##### DRUM DOI

doi:10.13016/M2CJ87M2J

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In this thesis we study the Deligne-Simpson problem of finding matrices $A_j\in C_j$ such that $A_1A_2\ldots A_k = I$ for $k\geq 3$ fixed loxodromic conjugacy classes $C_1,\ldots,C_k$ in $PU(n,1)$. Solutions to this problem are equivalent to representations of the $k$ punctured sphere into $PU(n,1)$, where the monodromy around the punctures are in the $C_j$. By Simpson's correspondence \cite{s1}, irreducible such representations correspond to stable parabolic $U(n,1)$-Higgs bundles of parabolic degree 0. A parabolic $U(n,1)$-Higgs bundle can be decomposed into a parabolic $U(1,1)$-Higgs bundle and a $U(n-1)$ bundle by quotienting out by the rank $n-1$ kernel of the Higgs field. In the case that the $U(1,1)$-Higgs bundle is of loxodromic type, this construction can be reversed, with the added consequence that the stability conditions of the resulting $U(n,1)$-Higgs bundle are determined only by the kernel of $\Phi$, the number of marked points, and the degree of the $U(1,1)$-Higgs bundle. With this result, we prove our main theorem, which says that when the log eigenvalues of lifts $\widetilde{C}_j$ of the $C_j$ to $U(n,1)$ satisfy the inequalities in \cite{biswas} for the existence of a stable parabolic bundle, then there is a stable parabolic $U(n,1)$-Higgs bundle whose monodromies around the marked points are in $\widetilde{C}_j$. This new approach using Higgs bundle techniques generalizes the result of Falbel and Wentworth in \cite{fw1} for fixed loxodromic conjugacy classes in $PU(2,1)$.
This new result gives sufficient, but not necessary, conditions for the existence of an irreducible solution to the Deligne-Simpson problem for fixed loxodromic conjugacy classes in $PU(n,1)$. The stability assumption cannot be dropped from our proof since no universal characterization of unstable bundles exists. In the last chapter, we explore what happens when we change the weights of the stable kernel in the special case of three fixed loxodromic conjugacy classes in $PU(3,1)$. Using the techniques from \cite{fw2}, \cite{fw1}, and \cite{paupert}, we can show that our construction implies the existence of many other solutions to the problem.

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