Three series of 3-manifolds and their fundamental groups (Q1363958)

The authors study the elementary characteristics of the fundamental groups of some compact closed oriented 3-manifolds. Their main results are contained in Theorems 1-3 below. Theorem 1. The factor group with respect to the commutant \(\overline G_k(n)= G_k(n)/G_k'(n)\); \(k= 1,2,3\), considered in additive notation, has the following structure: (1) \(\overline G_1(3d) \cong Z^{2d} \oplus Z_3^{d+2}\); \(\overline G_1(n) \cong Z_9 \oplus Z_3^n\); \(n\not \equiv 0\pmod 3\); (2) \(|\overline G_2(n) |< \infty\); (3) \(\overline G_3(n) \cong Z\oplus Z_2\); \(n\equiv 1\pmod 3\); \(\overline G_3(n) \cong Z\); \(n\not \equiv 1\pmod 3\). Here, \(Z_m= Z/mZ\), \(Z^n_3\) is the direct sum of \(n\) copies of the group \(Z_3\). (Proposition. The balanced groups \(G_1(n)\), \(n\geq 2\); \(G_2(n)\), \(n\geq 2\); \(G_3(n)\), \(n\geq 1\), from Sect. 1 are realized as the fundamental groups of the closed compact oriented 3-manifolds \(M^k_n\): \(G_k(n) \cong \pi_1 (M^k_n)\), \(k=1,2,3\).) Theorem 2. All the groups \(G_k(n)\) are infinite, except possibly group \(G_1(1)\). Theorem 3. For any \(n\equiv 2\pmod 3\) the group \(G_1(n)\) has an epimorphism to an infinite non-Abelian subgroup in \(SL_2(F)\), where \(F\) is the algebraic extension of order 6 of the field \(Q\).