We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.
{"title":"Flip graphs for infinite type surfaces","authors":"A. Fossas, H. Parlier","doi":"10.4171/ggd/685","DOIUrl":"https://doi.org/10.4171/ggd/685","url":null,"abstract":"We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48554114","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}
In this paper we show the statement in the title. To any Garside group of finite type, Wiest and the author associated a hyperbolic graph called the emph{additional length graph} and they used it to show that central quotients of Artin-Tits groups of spherical type are acylindrically hyperbolic. In general, a euclidean Artin-Tits group is not emph{a priori} a Garside group but McCammond and Sulway have shown that it embeds into an emph{infinite-type} Garside group which they call a emph{crystallographic Garside group}. We associate a emph{hyperbolic} additional length graph to this crystallographic Garside group and we exhibit elements of the euclidean Artin-Tits group which act loxodromically and WPD on this hyperbolic graph.
{"title":"Euclidean Artin–Tits groups are acylindrically hyperbolic","authors":"M. Calvez","doi":"10.4171/ggd/683","DOIUrl":"https://doi.org/10.4171/ggd/683","url":null,"abstract":"In this paper we show the statement in the title. To any Garside group of finite type, Wiest and the author associated a hyperbolic graph called the emph{additional length graph} and they used it to show that central quotients of Artin-Tits groups of spherical type are acylindrically hyperbolic. In general, a euclidean Artin-Tits group is not emph{a priori} a Garside group but McCammond and Sulway have shown that it embeds into an emph{infinite-type} Garside group which they call a emph{crystallographic Garside group}. We associate a emph{hyperbolic} additional length graph to this crystallographic Garside group and we exhibit elements of the euclidean Artin-Tits group which act loxodromically and WPD on this hyperbolic graph.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48779588","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 $G_n$ denote either $Aut(F_n)$, the automorphism group of a free group of rank $n$, or $Mod(Sigma_n^1)$, the mapping class group of an orientable surface of genus $n$ with $1$ boundary component. In both cases $G_n$ admits a natural filtration ${G_n(k)}_{k=1}^{infty}$ called the Johnson filtration. The first terms of this filtration $G_n(1)$ are the subgroup of $IA$-automorphisms and the Torelli subgroup, respectively. It was recently proved for both families of groups that for each $k$, the $k^{rm th}$ term $G_n(k)$ is finitely generated when $n>>k$; however, no information about finite generating sets was known for $k>1$. The main goal of this paper is to construct an explicit finite generating set for $[IA_n,IA_n]$, the second term of the Johnson filtration of $Aut(F_n)$, and an almost explicit finite generating set for the Johnson kernel, the second term of the Johnson filtration of $Mod(Sigma_n^1)$.
{"title":"Effective finite generation for $[mathrm{ IA}_n,mathrm{ IA}_n]$ and the Johnson kernel","authors":"M. Ershov, Daniel Franz","doi":"10.4171/GGD/727","DOIUrl":"https://doi.org/10.4171/GGD/727","url":null,"abstract":"Let $G_n$ denote either $Aut(F_n)$, the automorphism group of a free group of rank $n$, or $Mod(Sigma_n^1)$, the mapping class group of an orientable surface of genus $n$ with $1$ boundary component. In both cases $G_n$ admits a natural filtration ${G_n(k)}_{k=1}^{infty}$ called the Johnson filtration. The first terms of this filtration $G_n(1)$ are the subgroup of $IA$-automorphisms and the Torelli subgroup, respectively. It was recently proved for both families of groups that for each $k$, the $k^{rm th}$ term $G_n(k)$ is finitely generated when $n>>k$; however, no information about finite generating sets was known for $k>1$. The main goal of this paper is to construct an explicit finite generating set for $[IA_n,IA_n]$, the second term of the Johnson filtration of $Aut(F_n)$, and an almost explicit finite generating set for the Johnson kernel, the second term of the Johnson filtration of $Mod(Sigma_n^1)$.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43324881","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}
How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in "Asymptotic invariants of infinite groups", we define homological filling functions of groups with coefficients in a group $R$. Our main theorem is that the coefficients make a difference. That is, for every $n geq 1$ and every pair of coefficient groups $A, B in {mathbb{Z},mathbb{Q}} cup {mathbb{Z}/pmathbb{Z} : ptext{ prime}}$, there is a group whose filling functions for $n$-cycles with coefficients in $A$ and $B$ have different asymptotic behavior.
在无方向曲面的Cayley图中填充一个循环有多难?根据Gromov在“无穷群的渐近不变量”中的注释，我们定义了群中带系数群的同调填充函数$R$。我们的主要定理是系数是有区别的。即对于每一个$n geq 1$和每一对系数群$A, B in {mathbb{Z},mathbb{Q}} cup {mathbb{Z}/pmathbb{Z} : ptext{ prime}}$，都有一个群，其对$A$和$B$中有系数的$n$ -环的填充函数具有不同的渐近行为。
{"title":"Homological filling functions with coefficients","authors":"Xing-xiao Li, Fedor Manin","doi":"10.4171/ggd/675","DOIUrl":"https://doi.org/10.4171/ggd/675","url":null,"abstract":"How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in \"Asymptotic invariants of infinite groups\", we define homological filling functions of groups with coefficients in a group $R$. Our main theorem is that the coefficients make a difference. That is, for every $n geq 1$ and every pair of coefficient groups $A, B in {mathbb{Z},mathbb{Q}} cup {mathbb{Z}/pmathbb{Z} : ptext{ prime}}$, there is a group whose filling functions for $n$-cycles with coefficients in $A$ and $B$ have different asymptotic behavior.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47419109","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 abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.
{"title":"Abstract group actions of locally compact groups on CAT(0) spaces","authors":"Philip Moller, Olga Varghese","doi":"10.4171/ggd/677","DOIUrl":"https://doi.org/10.4171/ggd/677","url":null,"abstract":"We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47954215","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}
Given a finite generating set $A$ for a group $Gamma$, we study the map $W mapsto WA$ as a topological dynamical system -- a continuous self-map of the compact metrizable space of subsets of $Gamma$. If the set $A$ generates $Gamma$ as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when $Gamma= mathbb{Z}^d$ and $A subseteq mathbb{Z}^d$ a finite positively generating set containing the natural invertible extension of the map $W mapsto W+A$ is always topologically conjugate to the unique "north-south" dynamics on the Cantor set. In contrast to this, we show that various natural "geometric" properties of the finitely generated group $(Gamma,A)$ can be recovered from the dynamics of this map, in particular, the growth type and amenability of $Gamma$. When $Gamma = mathbb{Z}^d$, we show that the volume of the convex hull of the generating set $A$ is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group $Gamma$, related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.
{"title":"Iterated Minkowski sums, horoballs and north-south dynamics","authors":"Jeremias Epperlein, Tom Meyerovitch","doi":"10.4171/ggd/670","DOIUrl":"https://doi.org/10.4171/ggd/670","url":null,"abstract":"Given a finite generating set $A$ for a group $Gamma$, we study the map $W mapsto WA$ as a topological dynamical system -- a continuous self-map of the compact metrizable space of subsets of $Gamma$. If the set $A$ generates $Gamma$ as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when $Gamma= mathbb{Z}^d$ and $A subseteq mathbb{Z}^d$ a finite positively generating set containing the natural invertible extension of the map $W mapsto W+A$ is always topologically conjugate to the unique \"north-south\" dynamics on the Cantor set. In contrast to this, we show that various natural \"geometric\" properties of the finitely generated group $(Gamma,A)$ can be recovered from the dynamics of this map, in particular, the growth type and amenability of $Gamma$. When $Gamma = mathbb{Z}^d$, we show that the volume of the convex hull of the generating set $A$ is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group $Gamma$, related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45836548","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 a special class of geometric quantizations with "small" quantum errors, the quantum classical correspondence gives rise to an asymptotic projective representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups.
{"title":"Asymptotic representations of Hamiltonian diffeomorphisms and quantization","authors":"L. Charles, L. Polterovich","doi":"10.4171/ggd/696","DOIUrl":"https://doi.org/10.4171/ggd/696","url":null,"abstract":"We show that for a special class of geometric quantizations with \"small\" quantum errors, the quantum classical correspondence gives rise to an asymptotic projective representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46470210","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 explore the dual version of Gottschalk's conjecture recently introduced by Capobianco, Kari, and Taati, and the notion of dual surjunctivity in general. We show that dual surjunctive groups satisfy Kaplansky's direct finiteness conjecture for all fields of positive characteristic. By quantifying the notions of injectivity and post-surjectivity for cellular automata, we show that the image of the full topological shift under an injective cellular automaton is a subshift of finite type in a quantitative way. Moreover we show that dual surjunctive groups are closed under ultraproducts, under elementary equivalence, and under certain semidirect products (using the ideas of Arzhantseva and Gal for the latter); they form a closed subset in the space of marked groups, fully residually dual surjunctive groups are dual surjunctive, etc. We also consider dual surjunctive systems for more general dynamical systems, namely for certain expansive algebraic actions, employing results of Chung and Li.
{"title":"On Dual surjunctivity and applications","authors":"M. Doucha, Jakub Gismatullin","doi":"10.4171/ggd/681","DOIUrl":"https://doi.org/10.4171/ggd/681","url":null,"abstract":"We explore the dual version of Gottschalk's conjecture recently introduced by Capobianco, Kari, and Taati, and the notion of dual surjunctivity in general. We show that dual surjunctive groups satisfy Kaplansky's direct finiteness conjecture for all fields of positive characteristic. By quantifying the notions of injectivity and post-surjectivity for cellular automata, we show that the image of the full topological shift under an injective cellular automaton is a subshift of finite type in a quantitative way. Moreover we show that dual surjunctive groups are closed under ultraproducts, under elementary equivalence, and under certain semidirect products (using the ideas of Arzhantseva and Gal for the latter); they form a closed subset in the space of marked groups, fully residually dual surjunctive groups are dual surjunctive, etc. We also consider dual surjunctive systems for more general dynamical systems, namely for certain expansive algebraic actions, employing results of Chung and Li.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43732540","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 consider an exhaustion of the modular orbifold by compact subsurfaces and show that the growth rate, in terms of word length, of the reciprocal geodesics on such subsurfaces (so named low lying reciprocal geodesics) converge to the growth rate of the full set of reciprocal geodesics on the modular orbifold. We derive a similar result for the low lying geodesics and their growth rate convergence to the growth rate of the full set of closed geodesics.
{"title":"Combinatorial growth in the modular group","authors":"Ara Basmajian, R. Valli","doi":"10.4171/ggd/667","DOIUrl":"https://doi.org/10.4171/ggd/667","url":null,"abstract":"We consider an exhaustion of the modular orbifold by compact subsurfaces and show that the growth rate, in terms of word length, of the reciprocal geodesics on such subsurfaces (so named low lying reciprocal geodesics) converge to the growth rate of the full set of reciprocal geodesics on the modular orbifold. We derive a similar result for the low lying geodesics and their growth rate convergence to the growth rate of the full set of closed geodesics.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44243520","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 construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.
{"title":"Finitely generated groups acting uniformly properly on hyperbolic space","authors":"Robert P. Kropholler, V. Vankov","doi":"10.4171/ggd/659","DOIUrl":"https://doi.org/10.4171/ggd/659","url":null,"abstract":"We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43106611","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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