On stability of periodic billiard orbits in polyhedra

Suppose a group G is generated by elements a1, . . . , an. Then any element g 6= 1 of G is represented as a product g1g2 . . . gk, where each gi is a generator aj or an inverse a −1 j . The smallest k that allows such a representation is called the length of g. The length of the unit element is set to 0. Notice that the length depends on the set of generators. The group G generated by a1, . . . , an is called a free group with n generators (a1, . . . , an are called free generators) if for any group H and any h1, . . . , hn ∈ H there exists a unique homomorphism f : G → H such that f(ai) = hi, 1 ≤ i ≤ n. A nontrivial element g ∈ G is represented as a1 i1 a m2 i2 . . . al il , where l ≥ 1, 1 ≤ ij ≤ n and mj 6= 0 for 1 ≤ j ≤ l, and ij 6= ij+1 for 1 ≤ j < l. The group G is free if and only if such a representation is unique for any g 6= 1. For any φ ∈ [0, 2π) let Aφ =  cosφ − sinφ 0 sinφ cosφ 0 0 0 1  , Bφ =  1 0 0 0 cosφ − sinφ 0 sinφ cosφ  .