Elements in finite classical groups whose powers have large   1-Eigenspaces
  We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic. 
Alice C. Niemeyer and Cheryl E. Praeger
math.GR