Mahler measure, links and homology growth
By Daniel S. Silver, Susan G. Williams. 2000
74 citations recorded by the source corpus.
Abstract Let l be an oriented link of d components with nonzero Alexander polynomial Δ(u1,…,ud). Let Λ be a finite-index subgroup of H 1 (S 3 −l)≅ Z d , and let MΛ be the corresponding abelian cover of S3 branched along l. The growth rate of the order of the torsion subgroup of H1(MΛ), as a suitable measure of Λ approaches infinity, is equal to the Mahler measure of Δ.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9b13b272f96cef527fc56a1985f8de65fe4ab6e5
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In Geometry and Topology
- Stefano Vidussi
- Stefan Friedl
- Akshay Venkatesh
- Nicolas Bergeron
- P. Michel
- R. Munshi
- Nicolas Templier
- V. Blomer
In Mathematical Physics
- Stefano Vidussi
- Stefan Friedl
- Nicolas Bergeron
- Akshay Venkatesh
- P. Michel
- R. Munshi
- V. Blomer
- Nicolas Templier
Citations in the corpus
- A SURVEY OF TWISTED ALEXANDER POLYNOMIALS (2009, background): “sure is defined as m(p) = exp Z1 θ1=0 ... Z1 θn=0 log p e2πiθ1,...,e2πiθ1 dθ1 ...dθn. Note that the integral can be singular, but one can show that the integral always converges. It is known (cf. e.g. [SW02]) that an integral one variable polynomial palways satisfies m(p) ≥ 1, and it satisfies m(p) = 1”
- The asymptotic growth of torsion homology for arithmetic groups (2010, methodology): “dule. If n1 6= n2, there exists a finite index (congruence) subgroup Γ ⊂SL2(Z[i]) such that the homology groups H1(Γ;Sn 1,n 2) have a non-trivial torsion subgroup. 2. (Follows from Silver and Williams [56].) Let k⊂S3 be a hyperbolic knot whose Alexander polynomial has nontrivial Mahler measure (e.g. ”
- ALEXANDER GROUPS AND VIRTUAL LINKS (2001, unknown): “Keywords: virtual knot, Alexander group.”