Geodesic
Everywhere Dense Subgroups of One Group of Tree Automorphisms
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 356-367

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The well-known question of the existence, in the Grigorchuk 2-groups, of subgroups that are everywhere dense in the profinite topology is answered negatively. This is shown to be equivalent to the fact that these groups do not contain maximal subgroups of infinite index. 
@article{TM_2000_231_a13,
     author = {E. L. Pervova},
     title = {Everywhere {Dense} {Subgroups} of {One} {Group} of {Tree} {Automorphisms}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {356--367},
     publisher = {mathdoc},
     volume = {231},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2000_231_a13/}
}
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E. L. Pervova. Everywhere Dense Subgroups of One Group of Tree Automorphisms. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 356-367. http://geodesic.mathdoc.fr/item/TM_2000_231_a13/