### A - finite actions and their automorphism groups.

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We establish the following model-theoretic characterization of the fragment IΔ₀ + Exp + BΣ₁ of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment IΔ₀ of Peano arithmetic with induction limited to Δ₀-formulae). Theorem A. The following two conditions are equivalent for a countable model of the language of arithmetic: (a) satisfies IΔ₀ + BΣ₁ + Exp; (b) $={I}_{fix}\left(j\right)$ for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here ${I}_{fix}\left(j\right)$ is the...

We study solvability of equations of the form ${x}^{n}=g$ in the groups of order automorphisms of archimedean-complete totally ordered groups of rank 2. We determine exactly which automorphisms of the unique abelian such group have square roots, and we describe all automorphisms of the general ones.

We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ω-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation...

Let $A$ be finite dimensional $\mathbf{C}$-algebra which is a complete intersection, i.e. $A=\mathbf{C}[{X}_{1},...,{X}_{n}]/({f}_{1},...,{f}_{n})$ whith a regular sequences ${f}_{1},...,{f}_{n}$. Steve Halperin conjectured that the connected component of the automorphism group of such an algebra $A$ is solvable. We prove this in case $A$ is in addition graded and generated by elements of degree 1.