Szymon Głąb: Dense free subgroups of automorphism groups of homogeneous partially ordered sets

Tuesday, November 15, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Szymon Głąb (Łódź University of Technology)

Title: Dense free subgroups of automorphism groups of homogeneous partially ordered sets

Abstract:

Let $1 \le n \le\omega$. Let $A_n$ be a set of natural numbers less than $n$. Define $<$ on $A_n$ so that for no
$x, y \in A_n$ is $x<y$. Let $B_n = A_n \times\mathbb{Q}$ where $\mathbb{Q}$ is the set of rational numbers. Define $<$ on $B_n$ so
that $(k, p) < (m, q)$ iff $k = m$ and $p < q$. Let $C_n = B_n$ and define $<$ on $C_n$ so that $(k, p) < (m, q)$
iff $p < q$. Finally, let $(D, <)$ be the universal countable homogeneous partially ordered set, that is a
Fraisse limit of all finite partial orders.
A structure is called ultrahomogeneous, if every embedding of its finitely generated substructure
can be extended to an automorphism. Schmerl showed that there are only countably many,
up to isomorphism, ultrahomogeneous countable partially ordered sets. More precisely he proved the
following characterization:

Let $(H, <)$ be a countable partially ordered set. Then $(H, <)$ is ultrahomogeneous iff it
is isomorphic to one of the following:
$(A_n, <)$ for $1 \le n \le\omega$;
$(B_n, <)$ for $1 \le n \le\omega$;
$(C_n, <)$ for $1 \le n \le\omega$;
$(D, <)$.

Moreover, no two of the partially ordered sets listed above are isomorphic.
Consider automorphisms groups $Aut(A_\omega) = S_\infty$, $Aut(B_n)$, $Aut(C_n)$ and $Aut(D)$. We prove that
each of these groups contains two elements f, g such that the subgroup generated by f and g is free
and dense. By Schmerl’s Theorem the automorphism group of a countable infinite partially ordered
set is freely topologically 2-generated.

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