Spectral gap in the group of affine transformations over prime fields

We study random walks on the groups $\Bbb F^d_p \rtimes$ SL$_d$($\Bbb F_p$). We estimate the spectral gap in terms of the spectral gap of the projection to the linear part SL$_d$($\Bbb F_p$). This problem is motivated by an analogue in the group $\Bbb R^d \rtimes$ SO($d$), which have application to smoothness of self-similar measures.European Research Council (Advanced Research Grant ID: 267259), ISF (grant ID: 983/09), Simons Foundatio