Geodesic
Presentations of $m$-Complexity at Most 1 Defining the Trivial Group
Matematičeskie zametki, Tome 78 (2005) no. 3, pp. 349-357.

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A family of presentations of $m$-complexity at most 1 defining the trivial group and containing a $Q^{**}$-equivalent copy of every balanced presentation of the trivial group with $m$-complexity at most 1 is described. 
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S. G. Ivanov. Presentations of $m$-Complexity at Most 1 Defining the Trivial Group. Matematičeskie zametki, Tome 78 (2005) no. 3, pp. 349-357. http://geodesic.mathdoc.fr/item/MZM_2005_78_3_a2/

[1] Hog-Angeloni C., Metzler W., Sieradski A. J., Two-dimensional Homotopy and Combinatorial Group Theory, Cambridge University Press, Cambridge, 1993 | MR

[2] Ivanov S. G., “Codes of $m$-complexity 1”, Proc. Steklov. Institute Math. Suppl. 2, 2001, 61–70 | MR

[3] Kargapolov M. I., Merzlyakov Yu. I., Osnovy teorii grupp, Nauka, M., 1982 | MR | Zbl

[4] Sbrodova E. A., “Kody $m$-slozhnosti $2$”, Mezhdunarodnaya konferentsiya–shkola po geometrii i analizu, posvyaschennaya pamyati A. D. Aleksandrova, Tezisy dokl., Novosibirsk, 2002, 64–65