A geometric approach to Quillen's conjecture

We introduce admissible collections for a finite group G and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the Quillen dimension at p property, a strong version of Quillen's conjecture, at a given odd prime divisor p of jGj. Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen's conjecture for the p-groups complex, Ann. of Math. (2) 137 (1993), no. 3, 473-529], our techniques are simpler.