Groups Geometry and Dynamics最新文献

{"title":"Realizing invariant random subgroups as stabilizer distributions","authors":"Simon Thomas","doi":"10.4171/ggd/757","DOIUrl":"https://doi.org/10.4171/ggd/757","url":null,"abstract":". Suppose that ν is an ergodic IRS of a countable group G such that [ N G ( H ) : H ] = n < ∞ for ν -a.e. H ∈ Sub G . In this paper, we consider the question of whether ν can be realized as the stabilizer distribution of an ergodic action G ↷ ( X,µ ) on a standard Borel probability space such that the stabilizer map x (cid:55)→ G x is n -to-one.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"43 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135042423","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":"Group boundaries for semidirect products with $mathbb{Z}$","authors":"Craig R. Guilbault, Brendan Burns Healy, Brian Pietsch","doi":"10.4171/ggd/750","DOIUrl":"https://doi.org/10.4171/ggd/750","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192979","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":"Symbolic group varieties and dual surjunctivity","authors":"Xuan Kien Phung","doi":"10.4171/ggd/749","DOIUrl":"https://doi.org/10.4171/ggd/749","url":null,"abstract":"Let $G$ be a group. Let $X$ be an algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of rational points of $X$. We study algebraic group cellular automata $tau colon A^G to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M to X$ where $M$ is a finite memory. When $G$ is sofic and $K$ is uncountable, we show that if $tau$ is post-surjective then it is weakly pre-injective. Our result extends the dual version of Gottschalk's Conjecture for finite alphabets proposed by Capobianco, Kari, and Taati. When $G$ is amenable, we prove that if $tau$ is surjective then it is weakly pre-injective, and conversely, if $tau$ is pre-injective then it is surjective. Hence, we obtain a complete answer to a question of Gromov on the Garden of Eden theorem in the case of algebraic group cellular automata.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" 45","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135292799","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":"Constructing pseudo-Anosovs from expanding interval maps","authors":"Ethan Farber","doi":"10.4171/ggd/745","DOIUrl":"https://doi.org/10.4171/ggd/745","url":null,"abstract":"We investigate a phenomenon observed by W. Thurston wherein one constructs a pseudo-Anosov homeomorphism on the limit set of a certain lift of a piecewise-linear expanding interval map. We reconcile this construction with a special subclass of generalized pseudo-Anosovs, first defined by de Carvalho. From there we classify the circumstances under which this construction produces a pseudo-Anosov. As an application, we produce for each $g geq 1$ a pseudo-Anosov $phi_g$ on the surface of genus $g$ that preserves an algebraically primitive translation structure and whose dilatation $lambda_g$ is a Salem number.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"40 s3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135431079","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":"Splittings of triangle Artin groups","authors":"Kasia Jankiewicz","doi":"10.4171/ggd/740","DOIUrl":"https://doi.org/10.4171/ggd/740","url":null,"abstract":"We show that a triangle Artin group $mathrm{Art}_{MNP}$, where $Mleq Nleq P$, splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either $M>2$ or $N>3$. We also prove that all even 3-generator Artin groups are residually finite.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136183496","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":"Tight inclusions of $C^*$-dynamical systems","authors":"Yair Hartman, Mehrdad Kalantar","doi":"10.4171/ggd/739","DOIUrl":"https://doi.org/10.4171/ggd/739","url":null,"abstract":"We study a notion of tight inclusions of $C^$- and $W^$-dynamical systems, which is meant to capture a tension between topological and measurable rigidity of boundary actions. An important case of such inclusions is $C(X)subset L^infty(X, nu)$ for measurable boundaries with unique stationary compact models. We discuss the implications of this phenomenon in the description of Zimmer amenable intermediate factors. Furthermore, we prove applications in the problem of maximal injectivity of von Neumann algebras.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765929","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 continuity of the growth rate on the space of Coxeter systems","authors":"Tomoshige Yukita","doi":"10.4171/ggd/741","DOIUrl":"https://doi.org/10.4171/ggd/741","url":null,"abstract":"Floyd showed that if a sequence of compact hyperbolic Coxeter polygons converges, then so does the sequence of the growth rates of the Coxeter groups associated with the polygons. For the case of the hyperbolic 3-space, Kolpakov discovered the same phenomena for specific convergent sequences of hyperbolic Coxeter polyhedra. In this paper, we show that the growth rate is a continuous function on the space of Coxeter systems. This is an extension of the results due to Floyd and Kolpakov since the convergent sequences of Coxeter polyhedra give rise to that of Coxeter systems in the space of marked groups.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135197669","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":"Globally stable cylinders for hyperbolic CAT(0) cube complexes","authors":"Nir Lazarovich, Michah Sageev","doi":"10.4171/ggd/744","DOIUrl":"https://doi.org/10.4171/ggd/744","url":null,"abstract":"Rips and Sela introduced the notion of globally stable cylinders and asked if all Gromov hyperbolic groups admit such. We prove that hyperbolic cubulated groups admit globally stable cylinders.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302595","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 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":"6 1","pages":"0"},"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}

{"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":"40 1","pages":"0"},"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}