Intrinsic construction of $\mathrm{SL}_2(\mathbb Z_p)$ from $\mathrm{SL}_2(\mathbb Z)$

Question

We can construct the topological group of $p$-adic integers from the group of integers by taking an inverse limit: $$\mathbb Z_p = \varprojlim_{n > 0} (\mathbb Z / p^n \mathbb Z)$$ This construction "sees" only the group structure of $\mathbb Z$.

Is it possible to construct $\mathrm{SL}_2(\mathbb Z_p)$ from $\mathrm{SL}_2(\mathbb Z)$ "using" only the group structure of $\mathrm{SL}_2(\mathbb Z)$?

Answer 1

The reduction $SL_2(\Bbb{Z})\to \mathrm{SL}_2(\Bbb{Z}/p\Bbb{Z})$ is surjective with kernel $\Gamma(p^n)$ thus $$SL_2(\Bbb{Z}_p)\cong \varprojlim SL_2(\Bbb{Z}/p^n\Bbb{Z})\cong\varprojlim SL_2(\Bbb{Z})/ \Gamma(p^n)$$ where $\cong$ are given by the obvious maps.

Thus it reduces to the ways to define $\Gamma(p^n)$.