On algebraic $\sigma$-groups

We introduce the categories of algebraic $\sigma$-varieties and $\sigma$-groups over a difference field $(K,\sigma)$. Under a ``linearly $\sigma$-closed" assumption on $(K,\sigma)$ we prove an isotriviality theorem for $\sigma$-groups. This theorem immediately yields the key lemma in a proof of the Manin-Mumford conjecture. The present paper crucially uses ideas of Pilay and Ziegler (2003) but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski's work (2001) and are also closely related to papers of Pink and Roessler (2002 and 2004).