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We give a complete description of the size of the conjugacy classes of the automorphism group of the random graph with respect to Christensen's Haar null ideal. It is shown that every non-Haar null class contains a translated copy of a nonempty portion of every compact set and that there are continuum many non-Haar null conjugacy classes. Our methods also yield a new proof of an old result of Truss., Comment: Most of this work previously appeared in the first version of the paper arXiv:1705.07593 which was split into 3 articles

In order to understand the structure of the "typical" element of an automorphism group, one has to study how large the conjugacy classes of the group are. For the case when typical is meant in the sense of Baire category, Truss proved that there is a co-meagre conjugacy class in Aut(Q, <), the automorphism group of the rational numbers. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. We give a complete description of the size of the conjugacy classes of Aut(Q, <) with respect to this notion. In particular, we show that there exist continuum many non-Haar null conjugacy classes, illustrating that the random behaviour is quite different from the typical one in the sense of Baire category., Comment: Most of this work previously appeared in the first version of the paper arXiv:1705.07593 which was split into 3 articles, including arXiv:1808.06121

In order to understand the structure of the `typical' element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper, we generalize the theorems of Dougherty and Mycielski about $S_\infty$ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the Cofinal Strong Amalgamation Property. As an application we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set.

In order to understand the structure of the "typical" element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood, see e.g. the works of Glasner and Weiss, and Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behaviour of the homeomorphisms is entirely different from the generic case. For $\text{Homeo}^+([0,1])$ we describe the non-Haar null conjugacy classes and also show that their union is co-Haar null, for $\text{Homeo}^+(\mathbb{S}^1)$ we describe the non-Haar null conjugacy classes, and for $\mathcal{U}(\ell^2)$ we show that, apart from the classes of the multishifts, all conjugacy classes are Haar null. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set.

In this paper we classify some special reducts of the countable atomless Boolean algebra which we call functional reducts. We prove that there are exactly $13$ such structures up to first order interdefinability.

Let $\mathbb{F}_2^\omega$ denote the countably infinite dimensional vector space over the two element field and $\operatorname{GL}(\omega, 2)$ its automorphism group. Moreover, let $\operatorname{Sym}(\mathbb{F}_2^\omega)$ denote the symmetric group acting on the elements of $\mathbb{F}_2^\omega$. It is shown that there are exactly four closed subgroups, $G$, such that $\operatorname{GL}(\omega, 2)\leq G\leq \operatorname{Sym}(\mathbb{F}_2^\omega)$. As $\mathbb{F}_2^\omega$ is an $\omega$-categorical (and homogeneous) structure, these groups correspond to the first order definable reducts of $\mathbb{F}_2^\omega$. These reducts are also analyzed. In the last section the closed groups containing the infinite symplectic group $\operatorname{Sp}(\omega, 2)$ are classified.