Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userReversible Maps and Products of Involutions in Groups of IETS
67
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $6$.
rate research
Read More
We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, when the diffeomorphism is the composition of transversal linear involutions.
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 $operatorname{PGL}_2(mathbb{K})$.
We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.
We show that a relatively hyperbolic group quasi-isometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3-manifolds, we show that fundamental groups of closed 3-manifolds have linearly controlled asymptotic dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3-manifolds with non-empty boundary have asymptotic dimension 2.
A group $G$ is said to be periodic if for any $gin G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $mu$ is finite. Moreover if the group consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
