A solenoid \(X\) is a compact, finite-dimensional, connected abelian group with normalized Haar measure. Consider the action of a countable discrete group generated by continuous affine transformations on the solenoid. Let \(\text{Aut}(X)\) be the group of continuous automorphisms of \(X\), the group of affine transformations of \(X\) is a map defined as \(x\in X\to x_0\theta(x)\in X\) for some \(\theta\in\text{Aut}(X)\) and \(x_0\in X\), denoted by \(\text{Aff}(X)\). Given a group \(\Gamma\) and a homomorphism \(\Gamma\to\text{Aff}(X)\), there is a natural measure-preserving action \(\Gamma\curvearrowright(X, \mu)\). Let \(\pi_X\) be the corresponding Koopman representation of \(\Gamma\) on \(L^2(X, \mu)\). Recall that \(\Gamma\curvearrowright(X, \mu)\) is ergodic if and only if there is no non-zero invariant vector in the \(\pi_X (\Gamma)\)-invariant subspace \(L^2_0=(\mathbf{C 1}_X)^{\bot}\) of functions with zero mean. The action \(\Gamma\curvearrowright(X, \mu)\) is said to have the spectral gap property (or has a spectral gap) if there are no even almost invariant vectors in \(L^2_0 (X, \mu)\), that is, there is no sequence of unit vectors \(f_n\) in \(L^2_0(X, \mu)\) such that \(\lim_n\|\pi_X(\gamma)f_n-f_n\|= 0\) for all \(\gamma\in\Gamma\).
The paper is focused on the relationship between the spectral gap property and the strong ergodicity for groups of affine transformations of solenoids. It is proved that a measure-preserving group action does not have the spectral gap property if and only if there exists a \(p_a (\Gamma)\)-invariant proper subsolenoid \(Y\) of \(X\) such that the image of \(\Gamma\) in \(\text{Aff}(X/Y)\) is a virtually solvable group. The Pontrjagin dual group, the Fourier transformation, the Koopman representation of \(\text{Aff}(X)\) on \(L^2(X,\mu)\), and the group structure are used in the proof. Some examples are provided: a \(p\)-adic solenoid with a prime \(p\) and an \(a\)-adic solenoid with a positive integer \(a\).