### 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.

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