On torsion in finitely presented groups

Abstract

We give a uniform construction that, on input of a recursive presentation $P$ of a group, outputs a recursive presentation of a torsion-free group, isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain a known result, the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed. We apply our techniques to show that recognising embeddability of finitely presented groups is $\Pi^{0}_{2}$-hard, $\Sigma^{0}_{2}$-hard, and lies in $\Sigma^{0}_{3}$. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the $\Sigma^{0}_{2}$ sets which are closed under taking factors.Comment: 11 pages. This is the version submitted for publicatio

Connected components of compact matrix quantum groups and finiteness conditions

“…In [11], Chiodo and Vyas give an example of a finitely presented group for which N 1 \Γ is isomorphic to a non-trivial torsion (cyclic) group. This shows that, in general, Λ…”

“…Example 5.6. If G = C * (Γ), our construction reduces to the following construction of [11]. Set N 1 = Γ t and let N r , r > 1, be the (normal) subgroup generated by elements g ∈ Γ for which g n ∈ N r−1 for some n > 0.…”

“…Example 5.15. In [11] Chiodo and Vyas exhibited an example of a discrete group Γ dual to a cocommutative compact quantum group of torsion degree 2. Their methods can be generalized to show that the same holds for any free product Γ of (C m * C n ) and C ∞ , with amalgamation xy = z p , where m, n and p ≥ 2 are integers, and x, y, z denote generators, respectively, of the three cyclic groups.…”

“…This is due to the fact that the quotient of a group by the subgroup generated by the torsion subset may still contain non-trivial torsion elements. An example has been constructed by M. Chiodo in [11]. From the quantum group viewpoint, this quotient is the dual of a subgroup, which is still disconnected.…”

“…From the quantum group viewpoint, this quotient is the dual of a subgroup, which is still disconnected. However, a simple ordinary inductive procedure [11] yields a sequence of quotients converging to the universal torsion-free quotient of [9], whose length may be infinite or arbitrarily finite (see Example 5.16).…”