We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors. Moreover, one gives an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $forallexists$-elementary. We also prove that every elementary embedding of an equationally noetherian group into itself is an automorphism.
{"title":"Elementary subgroups of virtually free groups","authors":"Simon André","doi":"10.4171/ggd/638","DOIUrl":"https://doi.org/10.4171/ggd/638","url":null,"abstract":"We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors. Moreover, one gives an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $forallexists$-elementary. We also prove that every elementary embedding of an equationally noetherian group into itself is an automorphism.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512896","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 extend results on transitive self-similar abelian subgroups of the group of automorphisms Am of an m-ary tree Tm in [2], to the general case where the permutation group induced on the first level of the tree has s ≥ 1 orbits. We prove that such a group A embeds in a self-similar abelian group A which is also a maximal abelian subgroup of Am. The construction of A is based on the definition of a free monoid ∆ of rank s of partial diagonal monomorphisms of Am, which is used to determine the structure of CAm(A), the centralizer of A in Am. Indeed, we prove A ∗ = CAm(∆(A)) = ∆(B(A)), where B(A) denotes the product of the projections of A in its action on the different s orbits of maximal subtrees of Tm and bar denotes the topological closure. When A is a torsion self-similar abelian group, it is shown that it is necessarily of finite exponent. Moreover, we extend recent constructions of self-similar free abelian groups of infinite enumerable rank to examples of such groups which are also ∆-invariant for s = 2. Finally, we focus on self-similar cyclic groups of automorphisms of Tm and compute their centralizers when m = 4.
{"title":"Self-similar abelian groups and their centralizers","authors":"A. C. Dantas, Tulio M. G. Santos, S. Sidki","doi":"10.4171/ggd/710","DOIUrl":"https://doi.org/10.4171/ggd/710","url":null,"abstract":"We extend results on transitive self-similar abelian subgroups of the group of automorphisms Am of an m-ary tree Tm in [2], to the general case where the permutation group induced on the first level of the tree has s ≥ 1 orbits. We prove that such a group A embeds in a self-similar abelian group A which is also a maximal abelian subgroup of Am. The construction of A is based on the definition of a free monoid ∆ of rank s of partial diagonal monomorphisms of Am, which is used to determine the structure of CAm(A), the centralizer of A in Am. Indeed, we prove A ∗ = CAm(∆(A)) = ∆(B(A)), where B(A) denotes the product of the projections of A in its action on the different s orbits of maximal subtrees of Tm and bar denotes the topological closure. When A is a torsion self-similar abelian group, it is shown that it is necessarily of finite exponent. Moreover, we extend recent constructions of self-similar free abelian groups of infinite enumerable rank to examples of such groups which are also ∆-invariant for s = 2. Finally, we focus on self-similar cyclic groups of automorphisms of Tm and compute their centralizers when m = 4.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49460906","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":"Erratum to “On groups with $S^2$ Bowditch boundary”","authors":"Bena Tshishiku, G. Walsh","doi":"10.4171/ggd/625","DOIUrl":"https://doi.org/10.4171/ggd/625","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46911216","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":"A complex Euclidean reflection group with a non-positively curved complement complex","authors":"B. Coté, Jon McCammond","doi":"10.4171/ggd/620","DOIUrl":"https://doi.org/10.4171/ggd/620","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47005686","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}
Sam Hughes, Eduardo Mart'inez-Pedroza, Luis Jorge S'anchez Saldana
For a finitely generated group $G$ and collection of subgroups $mathcal{P}$ we prove that the relative Dehn function of a pair $(G,mathcal{P})$ is invariant under quasi-isometry of pairs. Along the way we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned off Cayley graphs. We also prove that for a cocompact simply connected combinatorial $G$-$2$-complex $X$ with finite edge stabilisers, the combinatorial Dehn function is well-defined if and only if the $1$-skeleton of $X$ is fine. We also show that if $H$ is a hyperbolically embedded subgroup of a finitely presented group $G$, then the relative Dehn function of the pair $(G, H)$ is well-defined. In the appendix, it is shown that show that the Baumslag-Solitar group $mathrm{BS}(k,l)$ has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither $k$ divides $l$ nor $l$ divides $k$.
{"title":"Quasi-isometry invariance of relative filling functions","authors":"Sam Hughes, Eduardo Mart'inez-Pedroza, Luis Jorge S'anchez Saldana","doi":"10.4171/ggd/737","DOIUrl":"https://doi.org/10.4171/ggd/737","url":null,"abstract":"For a finitely generated group $G$ and collection of subgroups $mathcal{P}$ we prove that the relative Dehn function of a pair $(G,mathcal{P})$ is invariant under quasi-isometry of pairs. Along the way we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned off Cayley graphs. We also prove that for a cocompact simply connected combinatorial $G$-$2$-complex $X$ with finite edge stabilisers, the combinatorial Dehn function is well-defined if and only if the $1$-skeleton of $X$ is fine. We also show that if $H$ is a hyperbolically embedded subgroup of a finitely presented group $G$, then the relative Dehn function of the pair $(G, H)$ is well-defined. In the appendix, it is shown that show that the Baumslag-Solitar group $mathrm{BS}(k,l)$ has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither $k$ divides $l$ nor $l$ divides $k$.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46780503","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 class of $Z^{d}$-substitutive subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of $Z^{d}$. We prove that any measurable factor map and even any homomorphism associated to a matrix commuting with the expansion matrix, induces a continuous one. We also get strong restrictions on the normalizer group, proving that any endomorphism is invertible, the normalizer group is virtually generated by the shift action and the quotient of the normalizer group by the automorphisms is restricted by the digit tile of the substitution.
{"title":"Homomorphisms between multidimensional constant-shape substitutions","authors":"C. Cabezas","doi":"10.4171/ggd/726","DOIUrl":"https://doi.org/10.4171/ggd/726","url":null,"abstract":"We study a class of $Z^{d}$-substitutive subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of $Z^{d}$. We prove that any measurable factor map and even any homomorphism associated to a matrix commuting with the expansion matrix, induces a continuous one. We also get strong restrictions on the normalizer group, proving that any endomorphism is invertible, the normalizer group is virtually generated by the shift action and the quotient of the normalizer group by the automorphisms is restricted by the digit tile of the substitution.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41774092","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}
This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenability, a strengthening of amenability introduced in the 70's, and we prove that uniform amenability indeed is detectable from the profinite completion.
{"title":"Amenability and profinite completions of finitely generated groups","authors":"Steffen Kionke, E. Schesler","doi":"10.4171/ggd/732","DOIUrl":"https://doi.org/10.4171/ggd/732","url":null,"abstract":"This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenability, a strengthening of amenability introduced in the 70's, and we prove that uniform amenability indeed is detectable from the profinite completion.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48709241","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 prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group with certain other virtually cyclic groups, e.g. the direct product of the integer group with any non-abelian finite simple group.
{"title":"On continuous orbit equivalence rigidity for virtually cyclic group actions","authors":"Yongle Jiang","doi":"10.4171/ggd/709","DOIUrl":"https://doi.org/10.4171/ggd/709","url":null,"abstract":"We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group with certain other virtually cyclic groups, e.g. the direct product of the integer group with any non-abelian finite simple group.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44558668","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":"Iterated monodromy groups of Chebyshev-like maps on $mathbb{C}^n$","authors":"Joshua P. Bowman","doi":"10.4171/GGD/609","DOIUrl":"https://doi.org/10.4171/GGD/609","url":null,"abstract":"Every affine Weyl group appears as the iterated monodromy group of a Chebyshev-like polynomial self-map of $mathbb{C}^n$.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47347508","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}
Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichm"{u}ller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichm"{u}ller space. We show for any pseudo-Anosov mapping class $f$, there exists a power $n$, such that the number of lattice points of the $f^n$ conjugacy class intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{frac{h}{2}R}$.
{"title":"Growth of pseudo-Anosov conjugacy classes in Teichmüller space","authors":"Jiawei Han","doi":"10.4171/ggd/724","DOIUrl":"https://doi.org/10.4171/ggd/724","url":null,"abstract":"Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichm\"{u}ller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichm\"{u}ller space. We show for any pseudo-Anosov mapping class $f$, there exists a power $n$, such that the number of lattice points of the $f^n$ conjugacy class intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{frac{h}{2}R}$.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48452091","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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