Root-class residuality of fundamental group of a finite graph of group

D. V. Goltsov

Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 64-71.

Résumé

Let $\mathcal{K}$ be an abstract class of groups. Suppose $\mathcal{K}$ contains at least a non trivial group. Then $\mathcal{K}$ is called a root-class if the following conditions are satisfied: 1. If $A \in \mathcal{K}$ and $B \leq A$, then $B \in \mathcal{K}$. 2. If $A \in \mathcal{K}$ and $B \in \mathcal{K}$, then $A\times B \in \mathcal{K}$. 3. If $1\leq C \leq B \leq A$ is a subnormal sequence and $A/B, B/C \in \mathcal{K}$, then there exists a normal subgroup $D$ in group $A$ such that $D \leq C$ and $A/D \in \mathcal{K}$. Group $G$ is root-class residual (or $\mathcal{K}$-residual), for a root-class $\mathcal{K}$ if, for every $1 \not = g \in G$, exists a homomorphism $\varphi $ of group $G$ onto a group of root-class $\mathcal{K}$ such that $g\varphi \not = 1$. Equivalently, group $G$ is $\mathcal{K}$-residual if, for every $1 \not = g \in G$, there exists a normal subgroup $N$ of $G$ such that $G/N \in \mathcal{K}$ and $g \not \in N$. The most investigated residual properties of groups are finite groups residuality (residual finiteness), $p$-finite groups residuality and soluble groups residuality. All there three classes of groups are root-classes. Therefore results about root-class residuality have safficiently enough general character. Let $\mathcal{K}$ be a root-class of finite groups. And let $G$ be a fundamental group of a finite graph of groups with finite edges groups. The necessary and sufficient condition of virtual $\mathcal{K}$-residuality for the group $G$ is obtained. Bibliography: 16 titles.

Détail

Keywords: root-class of finite groups, fundamental group of a finite graph of groups, virtual $\mathcal{K}$-residuality.

Bibliographie

[1] D. Azarov, “Residual properties of generalized free products with cyclic amalgamation”, Commun. in Algebra, 43:4 (2015), 1464–1471 | DOI | MR | Zbl

[2] Azarov D. N., “Residuallity by same classes of finite groups of generalized free products of groups with normal amalgamation”, Sibirskii Math. J., 56:2 (2015), 249–264 (Russian) | MR | Zbl

[3] Azarov D. N., “On residual finiteness of HNN-extension and of generalized free products of finite rank groups”, Sibirskii Math. J., 54:6 (2013), 1203–1215 (Russian) | MR | Zbl

[4] Azarov D. N., “On the virtually residuallity by finite p-groups of descending HNN-extensions”, Chebyshevskii Sb., 13:1 (2012), 9–19 (Russian) | MR

[5] Azarov D. N., “On residual finiteness of generalized free products of groups of finite rank”, Sibirskii Math. J., 54:3 (2013), 485–497 (Russian) | MR | Zbl

[6] Azarov D. N., “On the virtually residuallity by finite p-groups of Baumslag–Solitar groups”, Modeling and Analysis of Information Systems, 20:1 (2013), 116–123 (Russian) | MR

[7] Azarov D. N., “On the virtually residuallity by finite p-groups”, Chebyshevskii Sb., 11:3(35) (2010), 11–20 (Russian)

[8] Azarov D. N., “Residual properties of automorphism groups and split extension”, Izvestiya VUZov. Mathematics, 2015, no. 8, 3–13 (Russian)

[9] Azarov D. N., “The residuallity of solvable groups of finite rank by some classes of finite groups”, Izvestiya VUZov. Mathematics, 2014, no. 8, 18–29 (Russian) | MR

[10] Azarov D. N., “Some residual properties of soluble groups of finite rank”, Chebyshevskii Sb., 15:1(49) (2014), 7–18 (Russian) | MR

[11] Gruenberg K. W., “Residual properties of infinite solublegroups”, Proc. London Math. Soc., 7 (1957), 29–62 | DOI | MR | Zbl

[12] Azarov D. N., Tieudjo D., “On the residuallity of a free product with amalgamation by a root class of groups”, Nauch. trudy of Ivanovo State University. Mathematics, 5 (2002), 6–10 (Russian)

[13] Serre J.-P., Trees, Springer-Verlag, 1980 | MR | Zbl

[14] Baumslag G., “On the residual finiteness of generalized free products of nilpotent groups”, Trans. Amer. Math. Soc., 106:2 (1963), 193–209 | DOI | MR | Zbl

[15] Lyndon R. C., Schupp P. E., Combinatorial group theory, Springer-Verlag, Berlin–Heidelberg–New York, 1977 | MR | Zbl

[16] Karrass A., Solitar D., “Subgroups of HNN groups and groups with one defining relation”, Can. J. Math., 28 (1971), 627–643 | DOI | MR