On the Symmetric Square Large Sieve for ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ and the Prime Geodesic Theorem for ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathbb{H}}^3$

In this paper, we improve the error term in the prime geodesic theorem for the Picard manifold ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathbb{H}}^3$. Instead of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathbb{H}}^3$, we establish a spectral large sieve inequality for symmetric squares over ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$. This enables us to improve the bound $O(T^{3+2\mathrm{/}3+\epsilon })$ of Balkanova and Frolenkov into $O(T^{3+1\mathrm{/}2+\epsilon })$ for the second moment of symmetric square $L$-functions over ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathbb{H}}^3$. The basic idea is to enlarge the spherical family ${\mathrm{\Pi }}_c^0(T)$ of Maass cusp forms on ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathbb{H}}^3$ into the family ${\mathrm{\Pi }}_c(T,\sqrt{T})$ of cuspidal representations on ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[i])\mathrm{\backslash }{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$.