If $n$ is big enough comparing to $i$'s and $j$'s, and $i$'s and $j$'s are sufficiently different, then this group satisfies the small cancellation condition $C'(\lambda)$ with $\lambda\le 1/6$ (that is these words do not contain common subwords of length less $\gt 1/6$ of their length). This implies that the group is infinite, hyperbolic, and not virtually cyclic. This means that "generically" this group is infinite. By the way, the group in Bowditch's notes is a particular case of complexes of Baumslag-Solitar groups studied here.
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Generally this group If $n$ is infinite, hyperbolic, big enough comparing to $i$'s and not virtually cyclic$j$'s, and $i$'s and $j$'s are sufficiently different, by then this group satisfies the observation of Gromov (or even before Gromov) that these presentations generically satisfy small cancelation cancellation condition $C'(\lambda)$ with arbitrary small $\lambda$. See Gromov's paper "Hyperbolic groups" or Olshanskii's paper "Almost every group is hyperbolic." Internat. J. Algebra Comput. 2 \lambda\le 1/6$ (1992), no. 1, 1–17 or Champetier, Christophe, Propriétés statistiques des groupes de présentation finie. [Statistical properties that is these words do not contain common subwords of finitely presented groups], Adv. Math. 116 (1995), nolength less $\gt 1/6$ of their length). 2This implies that the group is infinite, 197–262hyperbolic, and not virtually cyclic. |
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