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Solving equations of length seven over torsion-free groups (2017)
Journal Article
Bibi, M., & Edjvet, M. (in press). Solving equations of length seven over torsion-free groups. Journal of Group Theory, 21(1), doi:10.1515/jgth-2017-0032. ISSN 1433-5883

Prishchepov [16] proved that all equations of length at most six over torsion-free groups are solvable. A different proof was given by Ivanov and Klyachko in [12]. This supports the conjecture stated by Levin [15] that any equation over a torsion-fre... Read More

The infinite Fibonacci groups and relative asphericity (2017)
Journal Article
Edjvet, M., & Juhasz, A. (2017). The infinite Fibonacci groups and relative asphericity. Transactions of the London Mathematical Society, 4(1), doi:10.1112/tlm3.12007. ISSN 2052-4986

We prove that the generalised Fibonacci group F (r, n) is infinite for (r, n) ∈ {(7 + 5k, 5), (8 + 5k, 5) : k ≥ 0}. This together with previously known results yields a complete classification of the finite F (r, n), a problem that has its origins in... Read More

On the asphericity of a family of positive relative group presentations (2017)
Journal Article
Aldwaik, S., & Edjvet, M. (in press). On the asphericity of a family of positive relative group presentations. Proceedings of the Edinburgh Mathematical Society, doi:10.1017/S0013091516000419. ISSN 0013-0915

Excluding four exceptional cases, the asphericity of the relative presentation P= ⟨G; x|xmgxh⟩ for m ≥ 2 is determined. If H = ⟨g; h⟩ ≤ G, then the exceptional cases occur when H is isomorphic to C5 or C6.

Asphericity of a length four relative presentation (2016)
Journal Article
Bin Ahmad, A. G., Al-Mulla, M. A., & Edjvet, M. (2016). Asphericity of a length four relative presentation. Journal of Algebra and Its Applications, 16(4), doi:10.1142/S0219498817500761. ISSN 0219-4988

We consider the relative group presentation P = < G, X | R > where X = { x \} and R = { xg_1 xg_2 xg_3 x^{-1} g_4 }. We show modulo a small number of exceptional cases exactly when P is aspherical. If the subgroup H of G is given by H = < g_1^{-1}... Read More