Define the growth function γ associated with a finitely generated group and a specified choice of generators {gl7 -, gp} for the group as follows (compare [9]). For each positive integer s let γ(s) be the number of distinct group elements which can be expressed as words of length < s in the specified generators and their inverses. (For example, if the group is free abelian of rank 2 with specif...

We show that the fundamental group of the complement of any irreducible tame torus sextics in P is isomorphic to Z2 ∗ Z3 except one class. The exceptional class has the configuration of the singularities {C3,9, 3A2} and the fundamental group is bigger than Z2 ∗ Z3. In fact, the Alexander polynomial is given by (t 2 − t + 1). For the proof, we first reduce the assertion to maximal curves and the...

The geometry of moduli spaces of complex abelian varieties and of compactifications of those moduli spaces has been the object of much study in the last few years. Given such a compactified moduli space it is natural to ask about the fundamental group of a resolution of singularities. This problem has been studied in some special cases, for instance in [K], [HK] and [HS]. It follows from the re...

Fundamental group is one of the most important topological invariants for general manifolds, which can be directly used as manifolds classification. In this work, we provide a series of practical and efficient algorithms to compute fundamental groups for general 3-manifolds based on CW cell decomposition. The input is a tetrahedral mesh, while the output is symbolic representation of its first ...

For any finitely presentable group G, we show the existence of an isolated complex surface singularity link which admits infinitely many exotic Stein fillings such that the fundamental group of each filling is isomorphic to G. We also provide an infinite family of closed exotic smooth four-manifolds with the fundamental groupG such that each member of the family admits a non-holomorphic Lefsche...

In this paper we present an example of two polarized K3 surfaces which are not Fundamental Group Equivalent (their fundamental groups of the complement of the branch curves are not isomorphic; denoted by FGE) but the fundamental groups of their related Galois covers are isomorphic. For each surface, we consider a generic projection to CP and a degenerations of the surface into a union of planes...

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This result may appear striking as the Novikov-Betti numbers carry “abelian” information about X. We refer to [4], [3] for the definition of the Novikov-Betti numbers; an explicit definition will also be given below in the proof of Theorem 1. An alternative description of bi(ξ) uses homology of complex flat line bundles. Consider the variety Vξ of all complex flat line bundles L over X having t...