By Frank Rimlinger

ISBN-10: 082182421X

ISBN-13: 9780821824214

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D Henceforth we assume that no two cyclically adjacent Pi,Pi + \ are both units. Case 3: For some Ga, Gp in the generating set, qieGaUHa, q2sQv, and q^eGpUHp. By case 2, we may assume q\eGa and q3e Gp, so qx and q^EQDPFor the sake of vividness, let a=qx, b=q3, and set u=

Consulting figure 2, we see that Va and V$ are the unique P-maximal vertices in distinct — -classes of P. Hence any fundamental group system for P must contain both Ga and G$. However, in U(P), a~x~l and b~x, so that the fundamental group system for U(P) is either { Ga } or { Gb }. Suppose we select { Ga }. The problem is that G$ is supposed to be amalgamated with Gb, but Gb was not picked for the generating set of U(P). In fact there is no way to pick fundamental group systems G^ and G^p) for P and U(P) respectively so that the groups in GP "line up" over those in Gjj^py • Nevertheless, if we conjugate Gb by x, then the fundamental groups line up in the larger group U(P).

By lemma 2, lP{X')=lP{Yf), since P'-length only depends on p. 4. Corollary: (Stallings) (P\Df) is a pregroup, and U(P')~U(F) in the natural way. 3. By definition, (P',D') is closed under multiplication and inversion. Invariance of the length of reduced words follows from corollary 3. Since F CP', P' generates U(F). • Recall that given an edge e eE, the edge group for e is Ee. By corollary 4, the following definition of F(G,7) is equivalent to that in Serre [1980, chap. 1].

### Pregroups and Bass-Serre Theory by Frank Rimlinger

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