Dmitri Shakhmatov

Zariski and (precompact) Markov topologies in free groups and their subgroups

Let G be a group. Pick x\notin G and consider the free product G\wprod \langle x\rangle of G with the cyclic group \langle x\rangle \cong \Z generated by x. For every g\in G, let \mathrm{ev}_g: G\wprod \langle x\rangle\to G be the unique homomorphism which sends x to g and coincides with the identity on G. For every word w\in G\wprod \langle x\rangle, the set E_w=\{g\in G: \mathrm{ev}_g(w)=1\} is the solution set in G of the equation w=1 with a single variable x and coefficients taken from G. (Here 1 denotes the identity of G.) The {\em Zariski} topology of G is the coarsest topology on G having all solution sets E_w for w\in G\wprod \langle x\rangle closed in this topology. The family of all subsets of G which are closed in every (precompact) Hausdorff group topology on G forms the family of closed subsets for a unique topology on G called its {\em (precompact) Markov} topology. Markov topology of a group is always finer than its Zariski topology. We review the known results related to Markov’s old problem of the
coincidence of these two topologies, and we solve it affirmatively for free groups. Furthermore, we prove that the precompact Markov topology on a non-commutative free group is strictly finer than its Markov topology. Finally, we prove that the Markov (Zariski) topology of a subgroup H of a free group G coincides with the subspace topology inherited by H from the Markov (Zariski) topology of G. The correspondent question for precompact Markov topologies remains open.

This is a joint work with Víctor Hugo Yãnez (Nanjing Normal University, Nanjing, China).