Abstract
We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic sense. We consider descriptions of two sorts: first, recursive presentations in terms of generators and relators, and second, algorithms for computing the group operation. For both classes of descriptions, we show that a large class of natural algebraic properties, Markov properties, are not recursively recognizable, indeed they are \(\Pi ^0_2\)-hard to detect in recursively presented groups and \(\Pi ^0_1\)-hard to detect in computable groups. These theorems suffice to give a sharp complexity measure for the detection problem of a number of typical group properties, for example, being abelian, torsion-free, orderable. Some properties, like being cyclic, nilpotent, or solvable, are much harder to detect, and we give sharp characterizations of the corresponding detection problems from a number of them. We give special attention to orderability properties, as this was a main motivation at the beginning of this project.
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Notes
For groups G and H and the left group action \(\rho \) of H on itself, the regular wreath product of G by H is the semidirect product \(G^H \rtimes H\) where \(G^H\) is the direct sum of |H|-many copies of G. The regular wreath product is denoted \(G \wr H\).
One can describe a bi-ordering via a Magnus expansion. To bi-order \(F_2=\langle a,b ~|~\rangle \), consider the ring of formal power series in non-commuting variables \(X_a\) and \(X_b\), \(\mathbb {Z}[X_a,X_b]\). The map induced by
$$\begin{aligned}\begin{array}{l} a \rightarrow 1+X_a \\ a^{-1} \rightarrow 1 - X_a + X_a^2 -X_a^3 +\cdots \\ b\rightarrow 1+ X_b \\ b^{-1} \rightarrow 1 - X_b + X_b^2 -X_b^3 +\cdots \end{array} \end{aligned}$$is an injective homomophism from \(F_2\) to \(\mathbb {Z}[X_a,X_b]\), the image of which is the multiplicative subgroup generated by \(F'_2=\{1+ p(X_a, X_b) ~|~\)each term in \(p(X_a, X_b)\) has degree at least 1.\(\}\). One can order \(\mathbb {Z}[X_a,X_b]\) by writing each power series in a standard form: write the terms in increasing degree, and within each degree, order the terms lexicographically according to subscripts. To compare two series, compare the coefficients of the first term on which they differ. The ordering inherited by the subgroup \(F'_2\) pulls back to an ordering on \(F_2\) via the isomorphism described above.
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Bilanovic, I., Chubb, J. & Roven, S. Detecting properties from descriptions of groups. Arch. Math. Logic 59, 293–312 (2020). https://doi.org/10.1007/s00153-019-00690-x
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DOI: https://doi.org/10.1007/s00153-019-00690-x
Keywords
- Decision problems
- Combinatorial group theory
- Computability theory
- Computable model theory
- Arithmetical hierarchy
- Orderable groups