# Infinite graphs that are strongly orbit minimal

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Simon Smith

City University of New York (City Tech)

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**Minisymposium:**
STRUCTURE AND PROPERTIES OF VERTEX-TRANSITIVE GRAPHS

**Content:**
We say that two permutation groups $A, B$ on the same set $X$ are {\em strongly orbit equivalent} if $A$ and $B$ have the same orbits on the power set $2^X$. Groups strongly orbit equivalent to finite symmetric groups were first considered by J. von Neumann and O. Morgenstern in 1947.
We call a group $B$ {\em strongly orbit minimal} if $B$ is not strongly orbit equivalent to any of its proper subgroups. When looking for families of strongly orbit equivalent groups, it suffices to consider only those groups which are strongly orbit minimal.
We say a graph is {\em strongly orbit minimal} if its automorphism group is strongly orbit minimal. For example, the countable random graph is strongly orbit minimal, but the countable complete graph is not.
Strong orbit minimality appears to be related to other notions of minimality, like primitivity and $2$-distinguishability. In this talk, I'll describe what is known about infinite graphs which are strongly orbit minimal; my talk will include a number of open problems and conjectures.