Cyclically presented Sieradski groups with even number of generators and three dimensional manifolds

The Sieradski groups are defined by the presentation $S(m)=\langle x_1,x_2,\dots,x_m\mid x_ix_{i+1}, i=1,\dots,m\rangle$, where all subscripts are taken by $\mod m$. The generalized Sieradski groups $S(m,p,q)$ are groups with $m$-cyclic presentation $G_m(w)$, where word $w$ has a special form depending on coprime integers $p$ and $q$. We study the problem if a given presentation is geometric, i.e. it corresponds to a spine of a closed orientable $3$-manifold. It was shown by Cavicchioli, Hegenbarth, and Kim that the generalized Sieradski group presentation $S(m,p,q)$ corresponds to a spine of some $3$-manifold which we denote as $M(m,p,q)$. Moreover, $M(m,p,q)$ are $m$-fold cyclic coverings of $S^3$ branched over the torus $(p,q)$-knot. Howie and Williams proved that $M(2n,3,2)$ are $n$-fold cyclic coverings of the lens space $L(3,1)$. A. Vesnin and T. Kozlovskaya established that $M(2n,5,2)$ are $n$-fold cyclic coverings of the lens space $L(5,1)$. In this paper, we consider generalized Sieradski manifolds $M(2n,7,2)$ $n\geqslant 1$. We prove that the $n$-cyclic presentations of their groups are geometric, i.e., correspond to spines of closed connected orientable $3$-manifolds. Moreover, manifolds $M(2n,7,2)$ are the $n$-fold cyclic coverings of the lens space $L(7,1)$. For the classification some of the constructed manifolds, we use the Recognizer computer program.