By H. H. Schaefer

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Mod k C 2/; where l and are defined as above. R; ; S /, and the weights r0 ; : : : ; rk and s0 ; : : : ; sk 1 satisfy Eqs. (1) and (2). 1 Using this results we can estimate the length of the elements in this subgroup. Furthermore, Burillo, Cleary, Stein and Taback gave an estimation of the length of elements in T in the number of carets in [2]. As a consequence of both results, we obtain: Proposition 3. Z/ is non distorted in Thompson’s group T . Corollary 4. Z/ as ˛ a subgroup of Thompson’s group T .

Let ‰ be an endomorphism of G D Zm Fn , with n 2. , det Q D ˙1). t u The hopficity of Zm and Fn together with this last proposition provide immediately the following results. Corollary 5. Zm Fn is hopfian and not cohopfian. t u 2) is an automorphism if Corollary 6. Z/. t u Now we have the ingredients to prove the main result of this note. Theorem 7. G; End G/ is solvable. Sketch of the proof. b; v/ 2 G, and have to decide whether there exists an automorphism (resp. monomorphism, endomorphism) of Zm Fn sending one to the other; and in the affirmative case, find one of them.

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