Virtually unipotent curves in some non-NPC graph manifolds

TL;DR

The paper investigates the properties of fundamental groups of certain graph manifolds, showing that under specific geometric conditions, these groups have representations with eigenvalues roots of unity, indicating non-linearity.

Contribution

It establishes a link between geometric structures of graph manifolds and the algebraic properties of their fundamental groups, providing new insights into their linearity and representation theory.

Findings

Existence of an essential curve with special eigenvalue properties

Fundamental groups do not admit faithful finite-dimensional unitary representations

Provides a new proof of non-linearity over fields of positive characteristic

Abstract

Let $M$ be a graph manifold containing a single JSJ torus $T$ and whose JSJ blocks are of the form $\Sigma \times S^1$, where $\Sigma$ is a compact orientable surface with boundary. We show that if $M$ does not admit a Riemannian metric of everywhere nonpositive sectional curvature, then there is an essential curve on $T$ such that any finite-dimensional linear representation of $\pi_1(M)$ maps an element representing that curve to a matrix all of whose eigenvalues are roots of $1$. In particular, this shows that $\pi_1(M)$ does not admit a faithful finite-dimensional unitary representation, and gives a new proof that $\pi_1(M)$ is not linear over any field of positive characteristic.