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As is well known, the group $PSL(2,\mathbb Z)$ is isomorphic to the free product $C_2 \ast C_3$ of cyclic groups of order $2$ and $3$. Call the generators of the cyclic groups $S$ and $T$.

Problem: Given a prime number $p$ and a natural number $n$, write a presentation of the quotient $PSL(2, \mathbb Z/p^n\mathbb Z)$ with the images of $S$ and $T$ as generators.

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3 Answers 3

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I usually use Sunday's presentation: see MR0311782. His T has order 2 but your S will be what he denotes ST.

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A method to do this for the group $\textrm{PSL}_2 (\mathbb{F}_{p^n})$ can be found in the papers by Glover and Sjerve:

Representing $PSl_2(p)$ on a Riemann surface of least genus, L'Enseignement Mathématique 31 (1985)

The genus of $PSl_2(q)$, Journal für die reine und angewandte Mathematik 380 (1987).

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The group $PSL_2(\mathbb{Z}/p^n)$ is the automorphisms group of the $(p+1)$ regular tree of depth $n$, where at level $m$ of the tree you have the points of $\mathbb{P}(\mathbb{Z}/p^m)$. The main benefit of this view, is that you can understand the relations at each level, and then move inductively to the next one.

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