Laurent Bartholdi, Danil Fialkovski, Sergei O. Ivanov
Let $F$ be a free group. We present for arbitrary $ginmathbb{N}$ a textsc{LogSpace} (and thus polynomial time) algorithm that determines whether a given $win F$ is a product of at most $g$ commutators; and more generally, an algorithm that determines, given $win F$, the minimal $g$ such that $w$ may be written as a product of $g$ commutators (and returns $infty$ if no such $g$ exists). This algorithm also returns words $x_1,y_1,dots,x_g,y_g$ such that $w=[x_1,y_1]dots[x_g,y_g]$. These algorithms are also efficient in practice. Using them, we produce the first example of a word in the free group whose commutator length decreases under taking a square. This disproves in a very strong sense a~conjecture by Bardakov.
{"title":"On commutator length in free groups","authors":"Laurent Bartholdi, Danil Fialkovski, Sergei O. Ivanov","doi":"10.4171/ggd/747","DOIUrl":"https://doi.org/10.4171/ggd/747","url":null,"abstract":"Let $F$ be a free group. We present for arbitrary $ginmathbb{N}$ a textsc{LogSpace} (and thus polynomial time) algorithm that determines whether a given $win F$ is a product of at most $g$ commutators; and more generally, an algorithm that determines, given $win F$, the minimal $g$ such that $w$ may be written as a product of $g$ commutators (and returns $infty$ if no such $g$ exists). This algorithm also returns words $x_1,y_1,dots,x_g,y_g$ such that $w=[x_1,y_1]dots[x_g,y_g]$. These algorithms are also efficient in practice. Using them, we produce the first example of a word in the free group whose commutator length decreases under taking a square. This disproves in a very strong sense a~conjecture by Bardakov.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135648461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tomasz Downarowicz, Piotr Oprocha, Mateusz Więcek, Guohua Zhang
The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure $nu$ on the collection $widetilde{mathcal O}$ of linear orders of type $mathbb Z$ on $G$, invariant under the natural action of $G$ on such orders. Multiorders exist on any countable amenable group (and only on such groups) and every multiorder has the Følner property, meaning that almost surely the order intervals starting at the unit form a Følner sequence. Every free measure-preserving $G$-action $(X,mu,G)$ has a multiorder $(widetilde{mathcal O},nu,G)$ as a factor and has the same orbits as the $mathbb Z$-action $(X,mu,S)$, where $S$ is the successor map determined by the multiorder factor. Moreover, the sub-sigma-algebra $Sigma_{widetilde{mathcal O}}$ associated with the multiorder factor is invariant under $S$, which makes the corresponding $mathbb Z$-action $(widetilde{mathcal O},nu,widetilde{S})$ a factor of $(X,mu,S)$. We prove that the entropy of any $G$-process generated by a finite partition of $X$, conditional with respect to $Sigma_{widetilde{mathcal O}}$, is preserved by the orbit equivalence with $(X,mu,S)$. Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to $h(mu,T,mathcal P)=H(mu,mathcal P|mathcal P^-)$ known for $mathbb Z$-actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss (2000). The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra $Sigma$, as soon as the “orbit change” is measurable with respect to $Sigma$. In our variant, we replace the measurability assumption by a simpler one: $Sigma$ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion, we provide a characterization of the Pinsker sigma-algebra of any $G$-process in terms of an appropriately defined remote past arising from a multiorder. The paper has an appendix in which we present an explicit construction of a particularly regular (uniformly Følner) multiorder based on an ordered dynamical tiling system of $G$.
{"title":"Multiorders in amenable group actions","authors":"Tomasz Downarowicz, Piotr Oprocha, Mateusz Więcek, Guohua Zhang","doi":"10.4171/ggd/738","DOIUrl":"https://doi.org/10.4171/ggd/738","url":null,"abstract":"The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure $nu$ on the collection $widetilde{mathcal O}$ of linear orders of type $mathbb Z$ on $G$, invariant under the natural action of $G$ on such orders. Multiorders exist on any countable amenable group (and only on such groups) and every multiorder has the Følner property, meaning that almost surely the order intervals starting at the unit form a Følner sequence. Every free measure-preserving $G$-action $(X,mu,G)$ has a multiorder $(widetilde{mathcal O},nu,G)$ as a factor and has the same orbits as the $mathbb Z$-action $(X,mu,S)$, where $S$ is the successor map determined by the multiorder factor. Moreover, the sub-sigma-algebra $Sigma_{widetilde{mathcal O}}$ associated with the multiorder factor is invariant under $S$, which makes the corresponding $mathbb Z$-action $(widetilde{mathcal O},nu,widetilde{S})$ a factor of $(X,mu,S)$. We prove that the entropy of any $G$-process generated by a finite partition of $X$, conditional with respect to $Sigma_{widetilde{mathcal O}}$, is preserved by the orbit equivalence with $(X,mu,S)$. Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to $h(mu,T,mathcal P)=H(mu,mathcal P|mathcal P^-)$ known for $mathbb Z$-actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss (2000). The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra $Sigma$, as soon as the “orbit change” is measurable with respect to $Sigma$. In our variant, we replace the measurability assumption by a simpler one: $Sigma$ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion, we provide a characterization of the Pinsker sigma-algebra of any $G$-process in terms of an appropriately defined remote past arising from a multiorder. The paper has an appendix in which we present an explicit construction of a particularly regular (uniformly Følner) multiorder based on an ordered dynamical tiling system of $G$.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135689805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine inclusions of $C^$-algebras of the form $A^H subseteq A rtimes_{r} G$, where $G$ and $H$ are groups acting on a unital simple $C^$-algebra $A$ by outer automorphisms and $H$ is finite. It follows from a theorem of Izumi that $A^H subseteq A$ is $C^$-irreducible, in the sense that all intermediate $C^$-algebras are simple. We show that $A^H subseteq A rtimes_{r} G$ is $C^$-irreducible for all $G$ and $H$ as above if and only if $G$ and $H$ have trivial intersection in the outer automorphisms of $A$, and we give a~Galois type classification of all intermediate $C^$-algebras in the case when $H$ is abelian and the two actions of $G$ and $H$ on $A$ commute. We illustrate these results with examples of outer group actions on the irrational rotation $C^$-algebras. We exhibit, among other examples, $C^$-irreducible inclusions of AF-algebras that have intermediate $C^$-algebras that are not AF-algebras; in fact, the irrational rotation $C^$-algebra appears as an intermediate $C^*$-algebra.
我们研究了形式为$A^H subseteq A r G$的$C^$-代数的包含，其中$G$和$H$是通过外自同构作用于一元简单$C^$-代数$A$的群，且$H$是有限的。由Izumi的定理可知，a ^H的子集a $是C^$-不可约的，即所有中间的C^$-代数都是简单的。证明了对于上述所有$G$和$H$，当且仅当$G$和$H$在$A$的外自同构中有平凡交时，$A^H $的子集$A r G$是$C^$-不可约的，并给出了在$H$是阿贝的情况下，所有中间$C^$-代数的$G$和$H$对$A$交换的两个作用下的$G$和$H$的$伽罗瓦类型分类。我们用无理数旋转$C^$-代数上的外群作用举例来说明这些结果。在其他例子中，我们展示了af -代数的$C^$-不可约包含，它们具有非af -代数的中间$C^$-代数;事实上，无理数旋转$C^$-代数表现为中间$C^*$-代数。
{"title":"Inclusions of $C^*$-algebras arising from fixed-point algebras","authors":"Siegfried Echterhoff, Mikael Rørdam","doi":"10.4171/ggd/743","DOIUrl":"https://doi.org/10.4171/ggd/743","url":null,"abstract":"We examine inclusions of $C^$-algebras of the form $A^H subseteq A rtimes_{r} G$, where $G$ and $H$ are groups acting on a unital simple $C^$-algebra $A$ by outer automorphisms and $H$ is finite. It follows from a theorem of Izumi that $A^H subseteq A$ is $C^$-irreducible, in the sense that all intermediate $C^$-algebras are simple. We show that $A^H subseteq A rtimes_{r} G$ is $C^$-irreducible for all $G$ and $H$ as above if and only if $G$ and $H$ have trivial intersection in the outer automorphisms of $A$, and we give a~Galois type classification of all intermediate $C^$-algebras in the case when $H$ is abelian and the two actions of $G$ and $H$ on $A$ commute. We illustrate these results with examples of outer group actions on the irrational rotation $C^$-algebras. We exhibit, among other examples, $C^$-irreducible inclusions of AF-algebras that have intermediate $C^$-algebras that are not AF-algebras; in fact, the irrational rotation $C^$-algebra appears as an intermediate $C^*$-algebra.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135790547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the geometry of a Picard modular group","authors":"M. Deraux","doi":"10.4171/ggd/734","DOIUrl":"https://doi.org/10.4171/ggd/734","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41980116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for any Polish group G and any countable normal subgroup Γ/G, the coset equivalence relation G/Γ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.
{"title":"Quotients by countable normal subgroups are hyperfinite","authors":"Joshua Frisch, Forte Shinko","doi":"10.4171/ggd/719","DOIUrl":"https://doi.org/10.4171/ggd/719","url":null,"abstract":"We show that for any Polish group G and any countable normal subgroup Γ/G, the coset equivalence relation G/Γ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42008339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-divergence in the space of lattices","authors":"Nicolas de Saxcé","doi":"10.4171/ggd/720","DOIUrl":"https://doi.org/10.4171/ggd/720","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49091722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Relativizing characterizations of Anosov subgroups, I (with an appendix by Gregory A. Soifer)","authors":"M. Kapovich, B. Leeb","doi":"10.4171/ggd/723","DOIUrl":"https://doi.org/10.4171/ggd/723","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49620714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotic property C of the wreath product $mathbb Z wr mathbb Z$","authors":"Jingming Zhu, Yan Wu","doi":"10.4171/ggd/711","DOIUrl":"https://doi.org/10.4171/ggd/711","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46436504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cocycle superrigidity for profinite actions of irreducible lattices","authors":"Daniel Drimbe, A. Ioana, J. Peterson","doi":"10.4171/ggd/700","DOIUrl":"https://doi.org/10.4171/ggd/700","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45558598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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