Galois representations ramified at one prime and with suitably large image

Let $p\ge 7$ be a prime and $n>1$ be a natural number. We show that there exist infinitely many Galois representations $\varrho :Gal(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})\to G{L}_n({\mathbb{Z}}_p)$ which are unramified outside $\{p,\mathrm{\infty }\}$ with large image. More precisely, the Galois representations constructed have image containing the kernel of the mod-$p^t$ reduction map $S{L}_n({\mathbb{Z}}_p)\to S{L}_n(\mathbb{Z}\mathrm{/}{p}^t\mathbb{Z})$, where $t:=8(n^2-n)(3+\lfloor lo{g}_p(2^n+1)\rfloor )+8$. The results are proven via a purely Galois theoretic lifting construction. When $p\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}4$, our results are conditional since in this case, we assume a very weak version of Vandiver’s conjecture.