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Register a new userThe geometry of involutions in ranked groups with a TI-subgroup
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We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture that this path leads to a new identification theorem for $operatorname{PGL}_2(mathbb{K})$.
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An element $f$ of a group $G$ is reversible if it is conjugated in $G$ to its own inverse; when the conjugating map is an involution, $f$ is called strongly reversible. We describe reversible maps in certain groups of interval exchange transformations namely $G_n simeq (mathbb S^1)^n rtimesmathcal S_n $, where $mathbb S^1$ is the circle and $mathcal S_n $ is the group of permutations of ${1,...,n}$. We first characterize strongly reversible maps, then we show that reversible elements are strongly reversible. As a corollary, we obtain that composites of involutions in $G_n$ are product of at most four involutions. We prove that any reversible Interval Exchange Transformation (IET) is reversible by a finite order element and then it is the product of two periodic IETs. In the course of proving this statement, we classify the free actions of $BS(1,-1)$ by IET and we extend this classification to free actions of finitely generated torsion free groups containing a copy of $mathbb Z^2$. We also give examples of faithful free actions of $BS(1,-1)$ and other groups containing reversible IETs. We show that periodic IETs are product of at most $2$ involutions. For IETs that are products of involutions, we show that such 3-IETs are periodic and then are product of at most $2$ involutions and we exhibit a family of non periodic 4-IETs for which we prove that this number is at least $3$ and at most $6$.
The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank, although the same questions can be asked about other classes of objects, for example, groups definable in $omega$-stable and $o$-minimal theories. In many cases, answers are not known even in the classical category of algebraic groups over algebraically closed fields.
We use hyperbolic towers to answer some model theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_0$, but that there is a finitely generated model which omits $p_0^{(2)}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.
We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group $UT_3(Z)$, the continuous Heisenberg group $UT_3(R)$, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.
In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
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