Since this is not purely group theoretical and not a complete answer, this maybe should be more of a comment, but since you mentioned simplicial complexes perhaps you should check out the following paper to get you started:

Haglund, Frédéric, and Daniel T. Wise. "Special cube complexes." Geometric and Functional Analysis 17.5 (2008): 1551-1620.

Which is concerned with the fundamental groups of certain square complexes (VH complexes whose 1-cells are divided into two classes, "horizontal" and "vertical", and the attaching maps of squares alternate v-h-v-h). For instance, they prove that any fundamental group of a compact virtually clean (clean means attaching maps are embeddings, and this implies that the group splits as a clean graph of groups, as studied in [1]) VH-complex is linear.

Although this isn't purely group theoretic, it is at least mostly presentation theoretic, and the result itself isn't too hard to apply if you have for instance a finitely presented group. In this case, there is often an easy algorithm to check whether such a group has a VH-subdivision. (For instance, there is an example of such a procedure outlined in my paper with Wise [2]). After you still have to check the virtually clean condition, which may or may not be so easy depending on what you are doing.

[1] Wise, Daniel T. "The residual finiteness of negatively curved polygons of finite groups." Inventiones mathematicae 149.3 (2002): 579-617.

[2] Polák and Wise, "A Note on VH Subdivisions", To appear.