Virtually unipotent curves in some non-NPC graph manifolds

Abstract

Let [Formula: see text] be a graph manifold containing a single JSJ torus [Formula: see text] and whose JSJ blocks are of the form [Formula: see text], where [Formula: see text] is a compact orientable surface with boundary. We show that if [Formula: see text] does not admit a Riemannian metric of everywhere nonpositive sectional curvature, then there is an essential curve on [Formula: see text] such that any finite-dimensional linear representation of [Formula: see text] maps an element representing that curve to a matrix all of whose eigenvalues are roots of [Formula: see text]. In particular, this shows that [Formula: see text] does not admit a faithful finite-dimensional unitary representation, and gives a new proof that [Formula: see text] is not linear over any field of positive characteristic.