By Alberto Conte

**Read Online or Download Algebraic Threefolds. Proc. conf. Varenna, 1981 PDF**

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**Extra info for Algebraic Threefolds. Proc. conf. Varenna, 1981**

**Example text**

18-19 we have seen ways to construct the fundamental group of a complex from the fundamental groups of subcomplexes. The group-theoretic analogue of this procedure consists of certain ways of building up groups from subgroups. This ultimately leads to the notion of the fundamental group of a graph of groups which we shall describe below. We begin, however, with a discussion of some important special cases. I. 1. Definition. Let (Gi)iEI be a family of groups and (K,i : A -> Gi)iEI a family of embeddings of the fixed group A in the various groups Gi .

Every countable group H can be embedded in a group G generated by two elements. Proof. Let H be a countable group and let the elements of H be enumerated as 1 = h o, hI, h 2 , .... Let Go be the free product H * F where F is the free group with basis {a, b}. Let Al = (b-nab n : n = 0,1,2, ... ) and A2 = (hna-nba n : n = 0,1,2, ... ). An easy cancellation argument shows that Al is free on the given generators (the generators actually form a Nielsen 1. Combinatorial Group Theory and Fundamental Groups 39 reduced system) and projecting from Go to F shows that the same holds for A 2.

Orphism f : 7r1 (C, v2-; Sym( n) there is an n-sheeted connected covering p : C -; C such that S (c, p) = (I). These results permit the effective determination of the n-sheeted coverings over a given complex C. We will explain it with an example. 12. Example. 6 (d). Then 7r1(N2) = (a, b I a2b2). The symmetric group Sym(3) consists of the six elements 1, r = (12), s = (13), t = (23), u = (123), v = (132), where r, s, t have order 2 and u, v have order 3. Elements f(a), f(b) with f(a)2 f(b)2 = 1 determine a homomorphism f : 7r1(N2) -; Sym(3).