## Isomorphism classes and derived series of certain almost-free groups.(English)Zbl 0837.20044

For a group $$G$$ denote by $$\gamma_n G$$ (correspondingly, by $$G^{(n)}$$) the $$n$$th member of its lower central series (derived series). Recall that $$\gamma_1 G=G=G^{(1)}$$, $$\gamma_{n+1} G=[\gamma_n G, G]$$ and $$G^{(n+1)}=[G^{(n)}, G^{(n)}]$$ for $$n\geq 1$$. G. Baumslag [Bull. Am. Math. Soc. 73, 621-622 (1967; Zbl 0153.35001)] introduced a class of groups $$G_{ij}=\langle a, b, c\mid a=[c^i,a]\cdot [c^j,b]\rangle$$; $$i,j\in \mathbb{Z}$$. He has proved that those groups, though not being free, closely resemble the free group $$F_2$$ on two generators. Namely, there hold $$G_{ij}/ \gamma_n G_{ij} \cong F_2/\gamma_n F_2$$ (for all $$n \in \mathbb{N})$$, and $$\bigcap_{n \geq 1} \gamma_n G_{ij}=(1)=\bigcap_{n \geq 1} \gamma_n F_2$$. It has been proved also that these groups $$G_{ij}$$ have the same (for all $$i, j \in \mathbb{Z}$$) quotients $$G_{ij}/\gamma_n G_{ij}$$ and $$G_{ij}/G^{(k)}_{ij}$$ for $$k=2, 3$$.
In this paper it is shown that there exist several distinct isomorphism types among the groups $$G_{ij}$$. Yet, it remains unknown whether all $$G_{ij}$$ are distinct or not. Also, it has been ‘computed’ by the authors that for some pairs $$(i,j)$$ the quotients $$G_{ij}/G^{(k)}_{ij}$$ for $$k \geq 4$$ can differ from the corresponding quotients for $$F_2$$. To obtain these results the authors have used computer-aided determination of the number of homomorphisms in the set $$\operatorname{Hom}(G_{ij}, \text{GL}(2, \mathbb{Z}_m))$$ for several pairs $$(i,j)$$ and $$m \in \mathbb{N}$$.

### MSC:

 20F14 Derived series, central series, and generalizations for groups 20F05 Generators, relations, and presentations of groups 20E34 General structure theorems for groups 20E05 Free nonabelian groups

Zbl 0153.35001
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### References:

  Baumslag G., Bull. Amer. Math. Soc. 73 pp 621– (1967) · Zbl 0153.35001  Havas G., Computational Group Theory: Durham, 1982 (1984)  Holt D., Groups, Combinatorics and Geometry: Durham, 1990 (1990)  Magnus W., Combinatorial Group Theory, (1976)
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