A surgery formula for the asymptotics of the higher dimensional Reidemeister torsion and Seifert fibered spaces

Abstract

We give a surgery formula for the asymptotic behavior of the sequence given by the logarithm of the higher dimensional Reidemeister torsion. Applying the resulting formula to Seifert fibered spaces, we show that the growth of the sequences has the same order as the indices and we give the explicit values for the limits of the leading coefficients. There are finitely many possibilities as the limits of the leading coefficients for a Seifert fibered space. We also show that the maximum is given by −χ log 2 where χ is the Euler characteristic of the base orbifold for a Seifert fibered space. These limits of the leading coefficients give a locally constant function on a character variety. This function takes the maximum −χ log 2 only on the top-dimensional components of the SU(2)-character varieties for Seifert fibered homology spheres.

Higher dimensional twisted Alexander polynomials for metabelian representations

“…It is easy seen that ψ j+p = ψ j for all j. This implies that {∆ K, ψ j (t)} j=1,2,... is a periodic sequence of rational functions in t with period p. Hence it follows from [Yam,Lemma 3.11] that…”

The Asymptotics of the Higher Dimensional Reidemeister Torsion for Exceptional Surgeries Along Twist Knots

“…By Proposition 4.1, the left hand side of (4) turns out to be The eigenvalues ξ ±1 k are primitive 2p k -th roots of unity as in Propositions 3.6 and 3.7. It follows from [Yam,Proposition 3.9] that the second term in the right hand side converges to −(log 2)/p k . Note that we can ignore the indeterminacy of a factor t j ( j ∈ Z) in the Alexander polynomial in the computation of the first term.…”

“…by the symmetry that ∆ K (t) = t j ∆ K (t −1 ) ( j ∈ Z) and [Yam,Lemma 3.11]. The Alexander polynomial ∆ T (2,2n+1) (t) is given by (t 2n+1 + 1)/(t + 1).…”

Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups

“…The author has studied the asymptotic behavior of the Reidemeister torsion for a Seifert fibered space and SL 2N (C)-representations of the fundamental group in [Yam17]. The observations in [Yam17] are based on explicit computations of the Reidemeister torsion in terms of the decomposition of a Seifert fibered space along tori.…”

“…The author has studied the asymptotic behavior of the Reidemeister torsion for a Seifert fibered space and SL 2N (C)-representations of the fundamental group in [Yam17]. The observations in [Yam17] are based on explicit computations of the Reidemeister torsion in terms of the decomposition of a Seifert fibered space along tori. The purpose of this paper is to study the relation between the Ruelle zeta function of a dynamical system and a sequence of the Reidemeister torsion for a Seifert fibered space M with SL 2N (C)representations of π 1 (M).…”