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Computing topological zeta functions of groups, algebras, and modules. II. (English) Zbl 1329.14053

The paper is a continuation of Part I [the author, Proc. Lond. Math. Soc. (3) 110, No. 5, 1099–1134 (2015; Zbl 1338.11082)].
Topological zeta functions of groups, algebras, and modules were introduced by M. du Sautoy and F. Loeser [Sel. Math., New Ser. 10, No. 2, 253–303 (2004; Zbl 1062.14029)] as asymptotic invariants related to the enumeration of subgroups, subalgebras and submodules respectively. The author gives a general algorithm making it possible, “in favorable circumstances”, to compute topological zeta functions. The author explains his method as follows. “At the heart of the present article lies the notion of a toric datum. A toric datum consists of a half-open cone within some Euclidean space and a finite collection of Laurent polynomials. We will begin our study of toric data in Section 3, where we will also relate them to the cone integral data of M. du Sautoy and F. Grunewald [Ann. Math. (2) 152, No. 3, 793–833 (2000; Zbl 1006.11051)]. As we will see, toric data give rise to associated \(p\)-adic integrals …and to topological zeta functions by means of a limit \(p\to 1\).”

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11M41 Other Dirichlet series and zeta functions
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
20E07 Subgroup theorems; subgroup growth
20F69 Asymptotic properties of groups
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References:

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