Let $G(4, 7)$ be an abstract group with the presentation $$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$ Richard Schwartz considered a representation of $G(4,7)$ into $PU(2,1)$ and extensively studied the image group denoted $\Gamma(4,7)$, for instance, the topology of the complex hyperbolic surface $H^{2}_{\mathbb{C}}/\Gamma(4, 7)$. But I'm more curious about $G(4,7)$ itself.
Is $G(4,7)$ a Coxeter group?
Whether $G(4,7)$ is a torsion group in which every element has finite order?
Edit: The second question has been answered in negative thanks to comments from Tom and Derek. I'd like to offer further context regarding the first question. Some topologists including me are interested in the end behavior of $G(4,7)$, for instance, is its boundary a Menger curve? Note that Jordane Granier in 2015 dissertation considered a similar group $$\langle \iota_1,\dots, \iota_6|\iota_{i}^{3} = id, \iota_{i}\iota_{i+1} = \iota_{i+1}\iota_{i}, i\in \mathbb{Z}/6\mathbb{Z} \rangle$$ It has been shown that this is a hyperbolic group with a Menger curve boundary. This was proved using the classic characterization of hyperbolic group boundaries by Kapovich-Kleiner. A pivotal aspect of Granier's proof involved geometrizing the group with the aid of Dymara-Osajda's building blocks. Given that $G(4,7)$ is hyperbolic, it seem natural to turn to Kapovich-Kleiner's characterization. Yet, an appropriate geometrization is necessary. This leads us to the first question, which, if answered affirmatively, would simplify the task. We do suspect it cannot be true, however, so far we don't know how to disprove it.
