95 research outputs found
Properties of the automorphism group and a probabilistic construction of a class of countable labeled structures
AbstractFor a class of countably infinite ultrahomogeneous structures that generalize edge-colored graphs we provide a probabilistic construction. Also, we give fairly general criteria for the automorphism group of such structures to have the small index property and strong uncountable cofinality, thus generalizing some results of Solecki, Rosendal, and several other authors
Hereditarily indecomposable continua as generic mathematical structures
We characterize the pseudo-arc as well as P-adic pseudo-solenoids (for a set
of primes P) as generic structures, arising from a natural game in which two
players alternate in building an inverse sequence of surjections. The second
player wins if the limit of this sequence is homeomorphic to a concrete (fixed
in advance) space, called generic whenever the second player has a winning
strategy.
For this aim, we develop a new approximate Fra\"iss\'e theory, in order to
realize the above-mentioned objects (the pseudo-arc and the pseudo-solenoids)
as Fra\"iss\'e limits. Our framework extends the discrete Fra\"iss\'e theory,
both classical and projective, and is also suitable for working directly with
continuous maps on metrizable compacta.
We show, in particular, that, when playing with continuous surjections
between non-degenerate Peano continua, the pseudo-arc is always generic. The
universal pseudo-solenoid appears to be generic over all surjections between
circle-like continua.Comment: 64 pages, 2 figures, comments are welcom
Topological properties of Wazewski dendrite groups
Homeomorphism groups of generalized Wa\.zewski dendrites act on the infinite
countable set of branch points of the dendrite and thus have a nice Polish
topology. In this paper, we study them in the light of this Polish topology.
The group of the universal Wa\.zewski dendrite is more
characteristic than the others because it is the unique one with a dense
conjugacy class. For this group , we show some of its topological
properties like existence of a comeager conjugacy class, the Steinhaus
property, automatic continuity and the small index subgroup property. Moreover,
we identify the universal minimal flow of . This allows us to prove
that point-stabilizers in are amenable and to describe the universal
Furstenberg boundary of .Comment: Slight modifications about the expositio
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