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Let $G$ be the modular group. We know this can be described by the relations (in terms of the $S$ and $T$ transformations) given by $S^4 = I, (ST)^3 = S^2$.

In my work matrix representations of $G$ have arisen where $T^N = I$ for some $N \geq 2$. I am interested to know whether the modular group $G$ with an additional relation $T^N = I$ gives a finite group? And regardless of this, what is known about $G$ with the additional relation $T^N = I$?

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    $\begingroup$ It is finite for $N \le 5$ and infinite for $N>5$ - see here for example. It is infinite euclidean for $N =6$ and infinite hyperbolic for $n \ge 7$. $\endgroup$
    – Derek Holt
    Commented Jul 2, 2022 at 15:46
  • $\begingroup$ @DerekHolt basically all my cases are in the double digits so I guess infinite hyperbolic. Can you provide any references that look at this case? $\endgroup$
    – JPhy
    Commented Jul 2, 2022 at 16:23

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