Polish Topologies for Graph Products of Groups

We give strong necessary conditions on the admissibility of a Polish group topology for an arbitrary graph product of groups $G(\Gamma, G_a)$, 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 $G_a$ are countable

Polish Topologies for Graph Products of Cyclic Groups

We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from [5], [7] and [4]

On strongly just infinite profinite branch groups

For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As an application, we show that many Burger-Mozes universal simple groups enjoy several automatic continuity properties.Comment: Typos and a minor error correcte

Bounded normal generation is not equivalent to topological bounded normal generation

We show that some derived $\mathrm{L}^1$ full groups provide examples of non simple Polish groups with the topological bounded normal generation property. In particular, it follows that there are Polish groups with the topological bounded normal generation property but not the bounded normal generation property.Comment: 11 page

Geometry of quantum dynamics in infinite dimension

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning coadjoint orbits of the unitary group in infinite dimension, embedding of the Hilbert projective space of pure states in the unitary group, and an approach to self-adjoint extensions of symmetric relations.Comment: 32 page