We consider the complexity class ACC¹ and related families of arithmetic circuits. We prove a variety of collapse results, showing several settings in which no loss of computational power results if fan-in of gates is severely restricted, as well as presenting a natural class of arithmetic circuits in which no expressive power is lost by severely restricting the algebraic degree of the circuits. We draw attention to the strong connections that exist between ACC¹ and VP, via connections to the classes CC¹[m] for various m. These results tend to support a conjecture regarding the computational power of the complexity class VP over finite algebras, and they also highlight the significance of a class of arithmetic circuits that is in some sense dual to VP. In particular, these dual-VP classes provide new characterizations of ACC¹ and TC¹ in terms of circuits of semiunbounded fan-in. As a corollary, we show that ACCⁱ = CCⁱ for all i ≥ 1.