. 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.
{"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":null,"pages":null},"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}
Craig R. Guilbault, Brendan Burns Healy, Brian Pietsch
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
For any order of growth $f(n)=o(log n)$ we construct a finitely-generated group $G$ and a set of generators $S$ such that the Cayley graph of $G$ with respect to $S$ supports a harmonic function with growth $f$ but does not support any harmonic function with slower growth. The construction uses permutational wreath products in which the base group is defined via its properly chosen Schreier graph.
{"title":"Groups with minimal harmonic functions as small as you like (with an appendix by Nicolás Matte Bon)","authors":"Gideon Amir, Gady Kozma","doi":"10.4171/ggd/748","DOIUrl":"https://doi.org/10.4171/ggd/748","url":null,"abstract":"For any order of growth $f(n)=o(log n)$ we construct a finitely-generated group $G$ and a set of generators $S$ such that the Cayley graph of $G$ with respect to $S$ supports a harmonic function with growth $f$ but does not support any harmonic function with slower growth. The construction uses permutational wreath products in which the base group is defined via its properly chosen Schreier graph.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726750","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}
Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set ${gin G | g^{p^n}=1}$ is a nontrivial subgroup for some $n$, then $G$ is a finite $p$-group with exponent at most $p^n$. This applies in particular to power abelian $p$-groups.
设$p$是一个素数，$G$是一个有限秩的亲$p$群，它在$p$任意根树上有忠实的、自相似的作用。证明了如果集合${g在g | g^{p^n}=1}$中是某$n$的非平凡子群，则$ g $是一个指数不超过$p^n$的有限$p$-群。这尤其适用于幂阿贝尔$p$-群。
{"title":"Exponent of self-similar finite $p$-groups","authors":"Alex Carrazedo Dantas, Emerson de Melo","doi":"10.4171/ggd/754","DOIUrl":"https://doi.org/10.4171/ggd/754","url":null,"abstract":"Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set ${gin G | g^{p^n}=1}$ is a nontrivial subgroup for some $n$, then $G$ is a finite $p$-group with exponent at most $p^n$. This applies in particular to power abelian $p$-groups.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018112","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 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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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