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Free quotients of finitely presented groups. (English) Zbl 0867.20030

Let \(G\) be a group with finite presentation \(\langle X\mid R\rangle\). The authors are concerned with the problem of determining whether or not \(G\) has a free (non-abelian) quotient. Any group which has a subgroup of finite index with such a quotient is known to be “large” (more precisely, \(SQ\)-universal).
Their approach to this problem is an interesting combination of computational and mathematical methods. The basic idea is to “add” relations \(R'\) to obtain a quotient of \(G\), \(\langle X\mid R\cup R'\rangle\) and then use Tietze transformations (with a computer) to verify that the latter group is free. The difficult question, of course, is choosing which relations to add.
After some “elementary” additions a computing algorithm is used to determine the maximal quotient \(\Gamma\) of the resulting group which lies in \({\mathcal N}_{2,p}\), the variety of all \(p\)-groups with exponent \(p\) class 2, for “low” \(p\). Mathematical methods are then used to determine relations that define a quotient of \(\Gamma\) which is free in \({\mathcal N}_{2,p}\). The latter relations are then “lifted” to \(G\) and added to \(R\), resulting in a group which is (hopefully) free.
The motivation for this paper arose from a question of the reviewer on the existence of low index congruence subgroups of certain Bianchi groups with a special type of free quotient. The authors apply their method successfully to the Picard group, \(\text{SL}_2(\mathbb{Z}[i])\).

MSC:

20F05 Generators, relations, and presentations of groups
20E05 Free nonabelian groups
20E07 Subgroup theorems; subgroup growth
20F18 Nilpotent groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)

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References:

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