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Title: Virtually unipotent elements of 3-manifold groups.

Abstract: Suppose a group G contains an infinite-order element g such that every finite-dimensional linear representation of G maps some nontrivial power of g to a unipotent matrix. Since unitary matrices are diagonalizable, and since a unipotent matrix is torsion if its entries lie in a field of positive characteristic, such a group G does not admit a faithful finite-dimensional unitary representation, nor is G linear over a field of positive characteristic. We discuss manifestations of the above phenomenon in various finitely generated groups, with an emphasis on 3-manifold groups.

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