Decomposition of the vertex group of 3-manifolds
A 4-graph \(G\) is a finite 4-regular graph with a proper 4-edge- coloration. For \(k\in\{0,1,2,3\}\) a \(k\)-residue in \(G\) is a connected component of a subgraph induced by all the edges of \(k\) pre-selected colors. A 3-gem (3-dimensional graph encoded manifold) is a 4-graph where the number of 0-residues (the vertices) plus the number of 3-residues equals the number of 2-residues. A 3-crystallization is a 3-gem \(G\) in which there are exactly 4 3-residues. Every closed 3-manifold can be represented by a crystallization [see \textit{M. Pezzana}, Atti Sem. Mat. Fis. Univ. Modena 23 (1974), 269-277 (1975; Zbl 0314.57005)]. Let \(G\) be a bipartite crystallization, thus representing a closed orientable 3- manifold \(M\). The colors of the edges are labelled by the numbers 0,1,2,3. The generators of the vertex group \(\xi(G)\) of \(G\) are a set of symbols in 1-1 correspondence with the vertex set of \(G\). Its relators are in 1-1 correspondence with the bicolored cycles of \(G\) (for details see the paper). The authors prove that the group \(\xi(M)\) (indeed \(\xi(G)\) is an invariant of \(M\)) is isomorphic to the free product of three copies of the fundamental group of \(M\) with \(\mathbb{Z}\). This result opens the possibility of searching for topologically invariant automorphisms of \(\xi\).
scientific article; zbMATH DE number 89517
scientific article