Finiteness of the group Q(3,1)

Determine whether the group \(Q(3,1)=T_3/\langle\langle u^2\rangle\rangle\), where \(u=x_0x_1x_0^{-1}x_1^{-2}\), is finite.

Background

The group $Q(3,1)$ is obtained from the group $T_3$ by quotienting by the normal closure of $u^2$. The paper establishes related structural information, including that the quotient obtained by imposing the squares of all generators is the finite dihedral group $D_7$.

However, imposing only $u^2=1$ does not yield a known finiteness conclusion. The authors describe computational and analogous evidence suggesting finiteness may be possible, while explicitly leaving the finiteness of $Q(3,1)$ unresolved.