L^2-torsion of free-by-cyclic groups (GGD/GEAR Seminar)

Matthew Clay (University of Arkansas)

Abstract: I will provide an upper bound on the $L^2$-torsion of a free-by-cyclic group, $-p^{(2)}(G_\Phi)$, in terms of a relative train-track representative for $\Phi$ in Aut(F). This result shares features with a theorem of Luck-Schick computing the $L^2$-torsion of the fundamental group of a 3-manifold that fibers over the circle in that it shows that the $L^2$-torsion is determined by the exponential dynamics of the monodromy. In light of the result of Luck-Schick, a special case of this bound is analogous to the bound on the volume of a 3-manifold that fibers over the circle with pseudo-Anosov monodromy by the normalized entropy recently demonstrated by Kojima-McShane.