On the positivity of twisted \(L^2\)-torsion for 3-manifolds (Q6596257)

Let \(N\) be a compact orientable irreducible \(3\)-manifold with empty or incompressible toral boundary. The \(L^2\)-torsion of \(N\) is a numerical topological invariant of \(N\). By \textit{W. Lück} [\(L^2\)-invariants: Theory and applications to geometry and \(K\)-theory. Berlin: Springer (2002; Zbl 1009.55001)], \(L^2\)-torsion of \(N\) can be expressed by the exponential of the simplicial volume of \(N.\) For a finitely generated free Hilbert \(\mathcal{N}(G)\)-chain complex \(C_{\ast}\) satisfying the condition that it is of finite length, weakly acyclic and of determinant class, the author defines \(L^2\)-torsion \(\tau^{(2)}(C_{\ast})\) of \(C_{\ast}\) as the alternating product of the Fuglede-Kadison determinants of the connecting homomorphisms. If either condition is not satisfied, the author defines the \(L^2\)-torsion to be \(0\) by convention. Let \(X\) be a finite CW complex with fundamental group \(G.\) By using admissible triples for higher-dimensional linear representations, the author gives a definition of \(L^2\)-chain complex of \(X\) twisted by \((G,\rho;\gamma)\) that is the Hilbert \(\mathcal{N}(H)\)-chain complex, where \(\gamma: G\rightarrow H\) is a homomorphism to a countable group \(H.\) Then the author expresses the twisted \(L^2\)-torsion of \(X\) twisted by \((G,\rho;\gamma)\) as the \(L^2\)-torsion of the \(L^2\)-chain complex of \(X.\) In this paper, the author proves that if \(N\) has infinite fundamental group, then the twisted \(L^2\)-torsion \(\tau^{(2)}(N,\rho)\) of \(N\) is strictly positive for any group homomorphism \(\rho: \pi_1(N)\rightarrow SL(n, \mathbb{C})\) and it is continuous when restricted to the subvariety of upper triangular representations.