Separability and efficiency under standard wreath product in terms of Cayley graphs. (Q1011032)

Before quoting the abstract of this paper we can remind of some terminology. The terms of standard wreath product, presentations, Cayley graphs are as usual. If \(B\) is a subgroup of \(G\), then \(G\) is said to be \(B\)-separable, if for each \(x\in G-B\), there exists \(N\triangleleft G\) with finite index such that \(x\not\in NB\). If \(\mathcal P=\langle\underline x,\underline r\rangle\) is a finite presentation of \(G\) then the Euler characteristic of \(\mathcal P\) is defined to be \(\mathcal X(\mathcal P)=1-|\underline x|+|\underline r|\) and \(\delta(G)=1-\text{rk}_\mathbb{Z}(H_1(G))+d(H_2(G))\), where \(\text{rk}_\mathbb{Z}(H_1(G))\) denotes the \(\mathbb{Z}\)-rank of the torsion free part of \(H_1(G) \) and \(d(H_2(G))\) the minimal number of generators of \(H_2(G)\). The group \(G\) is called efficient if \(\mathcal X=\delta(G)\). The abstract goes as follows: In this paper we are mainly interested in separability and efficiency under the standard wreath product. To do that we will first obtain a presentation, say \(\mathcal P_G\), for the standard wreath product in terms of Cayley graphs. Then we will prove our first main result of this paper, which can be thought of as an application of the result given in [\textit{J. W. Wamsley}, J. Algebra 27, 48-56 (1973; Zbl 0273.20031)] (or the general result in [\textit{A. S. Çevik}, Proc. Edinb. Math. Soc., II. Ser. 43, No. 2, 415-423 (2000; Zbl 0952.20021)]). Moreover, by considering the standard wreath product \(G\) of any finite groups \(B\) by \(A\), we will define the relationship between \(B\)-separability and efficiency, on \(G\), as another main result of this paper.