Any continuous action of SL(n,Z), where n > 2, on a r-dimensional mod 2 homology sphere factors through a finite group action if r < n - 1. In particular, any continuous action of SL(n+2,Z) on the n-dimensional sphere factors through a finite group action.
Parwani, Kamlesh, "Actions of SL(n,Z) on homology spheres" (2005). Faculty Research and Creative Activity. 7.