University of Arkansas Topology Seminar: 12/3/2015

Speaker: Matt Clay

Title: ℓ²−torsion of free-by-cyclic groups

I will provide an upper bound on the ℓ²−torsion of a free-by-cyclic group, −ρ(2)(𝔽 ⋊Φ ℤ), in terms of a relative train-track representative for Φ ∈ Aut(𝔽). This result shares features with a theorem of Lück−Schick computing the ℓ²−torsion of the fundamental group of a 3−manifold that fibers over the circle in that it shows that the ℓ²−torsion is determined by the exponential dynamics of the monodromy. In light of the result of Lück−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.