Let G be a semisimple algebraic group defined over , and let be a compact open subgroup of . We relate the asymptotic representation theory of and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both:We prove that there is a constant C, independent of G, such that the number of n-dimensional representations of grows slower than , confirming a conjecture of Larsen and Lubotzky. In fact, we can take . We also prove the same bounds for groups over local fields of large enough characteristic.
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