We consider the problem of when the ascending HNN extension of a free group
is residually finite. This is equivalent to the problem of when an
endomorphism of a free group can be extended to an automorphism of a
pro-finite closure of the free group group. We show that this problem can be
reduced to a problem about counting periodic points of polynomial maps on
algebraic varieties over fields of positive characteristics. Using some
results from algebraic geometry (the intersection theory, resolution of
singularities, etc.) we proved (together with Alexander Borisov) that, in
particular, every ascending HNN extension of the free group of rank 2 is
residually finite.
Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.