Polish topologies for graph products of groups

Abstract

We give strong necessary conditions on the admissibility of a Polish group topology for an arbitrary graph product of groups G(Γ,Ga), and use them to give a characterization modulo a finite set of nodes. As a corollary, we give a complete characterization in case all the factor groups Ga are countable.

Polish topologies for graph products of cyclic groups

“…(1) A word w in the alphabet Γ is a sequence (a α1 1 , ..., a 5) We say that the word w is reduced if there is no word with fewer syllables which spells the same element of G. (6) We say that the consecutive syllables a αi i and a αi+1 i+1 are adjacent if a i E Γ a i+1 . (7) We say that the word w is a normal form for g if it spells g and it is reduced.…”

“…In works in preparation we deal with the characterization problem faced here in the more general setting of graph products of general groups [6], and with questions of embeddability of graph products of groups into Polish groups [5].…”

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

“…Another important result in this direction is a theorem of Nikolov and Segal [45, Theorem 1.1], which says that every abstract homomorphism from a finitely generated in topological sense profinite group to any profinite group is continuous. Papers [14,17,20,23,39,43,49] deal with automatic continuity of abstract homomorphisms from locally compact Hausdorff groups to some discrete groups; papers [17,20] also deal with completely metrizable groups as domains.…”

Abstract homomorphisms from locally compact groups to discrete groups

“…not necessarily continuous) homomorphism ϕ : L → G automatically continuous? There are many results in this direction in the literature, see [11], [17], [21], [32] or [35]. In particular, Dudley [21] proved that every abstract homomorphism from a locally compact group to a free group is automatically continuous.…”