Szymon Głąb

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

Tuesday, November 15, 2016 17:15

Room: D1-215

Speaker:
Szymon Głąb

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:

1. $$(A_n, <)$$ for $$1 \le n \le\omega$$;

2. $$(B_n, <)$$ for $$1 \le n \le\omega$$;

3. $$(C_n, <)$$ for $$1 \le n \le\omega$$;

4. $$(D, <)$$.

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.