Infiniteness of a certain group defined by a presentation (Q1584106)

It is known [see \textit{A.~C.~Kim} and \textit{A.~I.~Kostrikin}, Dokl. Akad. Nauk, Ross. Akad. Nauk 340, No.~2, 158-160 (1995; Zbl 0907.57001); Mat. Sb. 188, No.~2, 3-24 (1997; Zbl 0906.20021)] that the group \(G_1(1)\) has the presentation \[ G_1(1)=\langle a,b\mid b^{-1}a^{-1}ba^{-1}b^{-1}=a^2b^2a^2,\;a^{-1}b^{-1}ab^{-1}a^{-1}=b^2a^2b^2\rangle. \] The main result of the paper is the proof of the following assertion. The group \(G_1(1)\) is infinite. The proof is carried out by means of the Reidemeister-Schreier method.