Approximating Novikov–Shubin numbers of virtually cyclic coverings

Abstract

We assign real numbers to finite sheeted coverings of compact CW complexes designed as finite counterparts to the Novikov-Shubin numbers. We prove an approximation theorem in the case of virtually cyclic fundamental groups employing methods from Diophantine approximation.(2) AA * : (ℓ 2 G) r → (ℓ 2 G) r given by x → xAA * . Here the matrix A * is obtained from A by transposing and applying the canonical involution ( λ g g) * = λ g g −1 to the entries. Let {E AA * λ } λ≥0 be the family of equivariant spectral projections obtained from r (2) AA * by Borel functional calculus, E AA * λ = χ [0,λ] (r (2) AA * ), where χ [0,λ] is the

The shrinkage type of knots

“…This lemma is a special case of [12,Proposition 12] and arguing similarly as there we obtain the first half of our desired result.…”

“…This lemma is a special case of [, Proposition 12] and arguing similarly as there we obtain the first half of our desired result. Proposition If has roots on the unit circle, then .…”

“…It follows that all singular values of S n and T n lie in [C −1/2 , C 1/2 ]. By standard singular value inequalities as in [11, 24.4.7(c)] (see also [12,Proposition 13]) we obtain C −1 σ n σ n C σ n .…”

“…Proof Compare [, p. 11]. We obtain the ‐chain complex of by applying the functor to the cellular chain complex of .…”

“…In Section 5 we give the proof of Theorem . As one important ingredient we cite a lemma from our earlier work . There we had studied a similar finite‐dimensional invariant, built out of the minimal singular value and its multiplicity, which more closely mimics the definition of Novikov–Shubin numbers.…”