Strong relative property $(T)$ and spectral gap of random walks

We consider strong relative property $(T)$ for pairs $(\Ga, G)$ where $\Ga$ acts on $G$. If $N$ is a connected Lie group and $\Ga$ is a group of automorphisms of $N$, we choose a finite index subgroup $\Ga ^0$ of $\Ga$ and obtain that $(\Ga, [\Ga ^0, N])$ has strong relative property $(T)$ provided Zariski-closure of $\Ga$ has no compact factor of positive dimension. We apply this to obtain the following: $G$ is a connected Lie group with solvable radical $R$ and a semisimple Levi subgroup $S$. If $S_{nc}$ denotes the product of noncompact simple factors of $S$ and $S_T$ denotes the product of simple factors in $S_{nc}$ that have property $(T)$, then we show that $(\Ga, R)$ has strong relative property $(T)$ for a Zariski-dense closed subgroup of $S_{nc}$ if and only if $R=[S_{nc},R]$. The case when $N$ is a vector group is discussed separately and some interesting results are proved. We also considered actions on solenoids $K$ and proved that if $\Ga$ acts on a solenoid $K$, then $(\Ga, K)$ has strong relative property $(T)$ under certain conditions on $\Ga$. For actions on solenoids we provided some alternatives in terms of amenability and strong relative property $(T)$. We also provide some applications to the spectral gap of $\pi (\mu)=\int \pi (g) d\mu (g)$ where $\pi$ is a certain unitary representation and $\mu$ is a probability measure.