This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions that the quotient complex is $delta$-hyperbolic. Additionally, under certain circumstances, we show that if the original action on the projection complex was a non-elementary WPD action, then so is the action of the quotient group on the quotient of the projection complex. This implies that the quotient group is acylindrically hyperbolic.
{"title":"Hyperbolic quotients of projection complexes","authors":"Matt Clay, J. Mangahas","doi":"10.4171/ggd/646","DOIUrl":"https://doi.org/10.4171/ggd/646","url":null,"abstract":"This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions that the quotient complex is $delta$-hyperbolic. Additionally, under certain circumstances, we show that if the original action on the projection complex was a non-elementary WPD action, then so is the action of the quotient group on the quotient of the projection complex. This implies that the quotient group is acylindrically hyperbolic.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46051947","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 the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite dimensional groups like $mathcal{S}_infty$, the group of all permutations of a countable set on the other side. Our main results are Automatic continuity (any homomorphism to a separable group is continuous), minimality of the Polish topology, identification of its universal Furstenberg boundary as the closed unit ball of a separable Hilbert space with its weak topology, identification of its universal minimal flow as the completion of some suspension of the action of the additive group of the reals on its universal minimal flow. All along the text, we lead a parallel study with the sibling group of isometries of a separable Hilbert space.
{"title":"The Polish topology of the isometry group of the infinite dimensional hyperbolic space","authors":"Bruno Duchesne","doi":"10.4171/GGD/713","DOIUrl":"https://doi.org/10.4171/GGD/713","url":null,"abstract":"We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite dimensional groups like $mathcal{S}_infty$, the group of all permutations of a countable set on the other side. Our main results are \u0000Automatic continuity (any homomorphism to a separable group is continuous), minimality of the Polish topology, identification of its universal Furstenberg boundary as the closed unit ball of a separable Hilbert space with its weak topology, identification of its universal minimal flow as the completion of some suspension of the action of the additive group of the reals on its universal minimal flow. \u0000All along the text, we lead a parallel study with the sibling group of isometries of a separable Hilbert space.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46570595","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 give the topological obstructions to be a leaf in a minimal lamination by hyperbolic surfaces whose generic leaf is homeomorphic to a Cantor tree. Then, we show that all allowed topological types can be simultaneously embedded in the same lamination. This result, together with results of Alvarez-Brum-Martinez-Potrie and Blanc, complete the panorama of understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations when the topology of the generic leaf is given. In all cases, all possible topologies can be realized simultaneously.
{"title":"Topology of leaves for minimal laminations by non-simply-connected hyperbolic surfaces","authors":"S. Alvarez, J. Brum","doi":"10.4171/ggd/645","DOIUrl":"https://doi.org/10.4171/ggd/645","url":null,"abstract":"We give the topological obstructions to be a leaf in a minimal lamination by hyperbolic surfaces whose generic leaf is homeomorphic to a Cantor tree. Then, we show that all allowed topological types can be simultaneously embedded in the same lamination. This result, together with results of Alvarez-Brum-Martinez-Potrie and Blanc, complete the panorama of understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations when the topology of the generic leaf is given. In all cases, all possible topologies can be realized simultaneously.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41247337","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}
It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, $text{Aut}(F_n)$, has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of $text{Aut}(F_n)$ for $n geq 4$ where $k$th powers of all transvections act trivially for some fixed $k$. Thus, for some values of $k$, we deduce that the quotient of $text{Aut}(F_n)$ by the subgroup generated by $k$th powers of transvections contains nonabelian free groups. This expands on results of Malestein and Putman and of Bridson and Vogtmann.
{"title":"Arithmetic quotients of the automorphism group of a right-angled Artin group","authors":"Justin Malestein","doi":"10.4171/ggd/691","DOIUrl":"https://doi.org/10.4171/ggd/691","url":null,"abstract":"It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, $text{Aut}(F_n)$, has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of $text{Aut}(F_n)$ for $n geq 4$ where $k$th powers of all transvections act trivially for some fixed $k$. Thus, for some values of $k$, we deduce that the quotient of $text{Aut}(F_n)$ by the subgroup generated by $k$th powers of transvections contains nonabelian free groups. This expands on results of Malestein and Putman and of Bridson and Vogtmann.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46922416","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}
Consider a subshift over a finite alphabet, $Xsubset Lambda^{mathbb Z}$ (or $XsubsetLambda^{mathbb N_0}$). With each finite block $BinLambda^k$ appearing in $X$ we associate the empirical measure ascribing to every block $CinLambda^l$ the frequency of occurrences of $C$ in $B$. By comparing the values ascribed to blocks $C$ we define a metric on the combined space of blocks $B$ and probability measures $mu$ on $X$, whose restriction to the space of measures is compatible with the weak-$star$ topology. Next, in this combined metric space we fix an open set $mathcal U$ containing all ergodic measures, and we say that a block $B$ is "ergodic" if $Binmathcal U$. In this paper we prove the following main result: Given $varepsilon>0$, every $xin X$ decomposes as a concatenation of blocks of bounded lengths in such a way that, after ignoring a set $M$ of coordinates of upper Banach density smaller than $varepsilon$, all blocks in the decomposition are ergodic. The second main result concerns subshifts whose set of ergodic measures is closed. We show that, in this case, no matter how $xin X$ is partitioned into blocks (as long as their lengths are sufficiently large and bounded), after ignoring a set $M$ of upper Banach density smaller than $varepsilon$, all blocks in the decomposition are ergodic. The first half of the paper is concluded by examples showing, among other things, that the small set $M$, in both main theorems, cannot be avoided. The second half of the paper is devoted to generalizing the two main results described above to subshifts $XsubsetLambda^G$ with the action of a countable amenable group $G$. The role of long blocks is played by blocks whose domains are members of a Folner sequence while the decomposition of $xin X$ into blocks (of which majority is ergodic) is obtained with the help of a congruent system of tilings.
{"title":"Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures","authors":"T. Downarowicz, M. Wikecek","doi":"10.4171/ggd/679","DOIUrl":"https://doi.org/10.4171/ggd/679","url":null,"abstract":"Consider a subshift over a finite alphabet, $Xsubset Lambda^{mathbb Z}$ (or $XsubsetLambda^{mathbb N_0}$). With each finite block $BinLambda^k$ appearing in $X$ we associate the empirical measure ascribing to every block $CinLambda^l$ the frequency of occurrences of $C$ in $B$. By comparing the values ascribed to blocks $C$ we define a metric on the combined space of blocks $B$ and probability measures $mu$ on $X$, whose restriction to the space of measures is compatible with the weak-$star$ topology. Next, in this combined metric space we fix an open set $mathcal U$ containing all ergodic measures, and we say that a block $B$ is \"ergodic\" if $Binmathcal U$. \u0000In this paper we prove the following main result: Given $varepsilon>0$, every $xin X$ decomposes as a concatenation of blocks of bounded lengths in such a way that, after ignoring a set $M$ of coordinates of upper Banach density smaller than $varepsilon$, all blocks in the decomposition are ergodic. The second main result concerns subshifts whose set of ergodic measures is closed. We show that, in this case, no matter how $xin X$ is partitioned into blocks (as long as their lengths are sufficiently large and bounded), after ignoring a set $M$ of upper Banach density smaller than $varepsilon$, all blocks in the decomposition are ergodic. The first half of the paper is concluded by examples showing, among other things, that the small set $M$, in both main theorems, cannot be avoided. \u0000The second half of the paper is devoted to generalizing the two main results described above to subshifts $XsubsetLambda^G$ with the action of a countable amenable group $G$. The role of long blocks is played by blocks whose domains are members of a Folner sequence while the decomposition of $xin X$ into blocks (of which majority is ergodic) is obtained with the help of a congruent system of tilings.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45844914","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 Borel graphs which settle several questions in descriptive graph combinatorics. These include "Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?" and "Can marked groups with isomorphic Cayley graphs have Borel chromatic numbers for their shift graphs which differ by more than one?" We also provide a new bound for Borel chromatic numbers of graphs whose connected components all have two ends.
{"title":"Descriptive chromatic numbers of locally finite and everywhere two-ended graphs","authors":"F. Weilacher","doi":"10.4171/ggd/643","DOIUrl":"https://doi.org/10.4171/ggd/643","url":null,"abstract":"We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include \"Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?\" and \"Can marked groups with isomorphic Cayley graphs have Borel chromatic numbers for their shift graphs which differ by more than one?\" We also provide a new bound for Borel chromatic numbers of graphs whose connected components all have two ends.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43850755","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 $M$ be a manifold, $N$ a 1-dimensional manifold. Assuming $r neq dim(M)+1$, we show that any nontrivial homomorphism $rho: text{Diff}^r_c(M)to text{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are countably many embeddings $phi_i: Mto N$ with disjoint images such that the action of $rho$ is conjugate (via the product of the $phi_i$) to the diagonal action of $text{Diff}^r_c(M)$ on $M times M times ...$ on $bigcup_i phi_i(M)$, and trivial elsewhere. This solves a conjecture of Matsumoto. We also show that the groups $text{Diff}^r_c(M)$ have no countable index subgroups.
{"title":"There are no exotic actions of diffeomorphism groups on 1-manifolds","authors":"Lei Chen, Kathryn Mann","doi":"10.4171/ggd/658","DOIUrl":"https://doi.org/10.4171/ggd/658","url":null,"abstract":"Let $M$ be a manifold, $N$ a 1-dimensional manifold. Assuming $r neq dim(M)+1$, we show that any nontrivial homomorphism $rho: text{Diff}^r_c(M)to text{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are countably many embeddings $phi_i: Mto N$ with disjoint images such that the action of $rho$ is conjugate (via the product of the $phi_i$) to the diagonal action of $text{Diff}^r_c(M)$ on $M times M times ...$ on $bigcup_i phi_i(M)$, and trivial elsewhere. This solves a conjecture of Matsumoto. We also show that the groups $text{Diff}^r_c(M)$ have no countable index subgroups.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42536315","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 PC be the group of bijections from [0, 1[ to itself which are continuous outside a finite set. Let PC be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of PC vanishes. That is, the quotient map PC $rightarrow$ PC splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, from PC to Z 2Z. Then we use this signature to list normal subgroups of every subgroup G of PC which contains S fin and such that G, the projection of G in PC , is simple.
{"title":"Signature for piecewise continuous groups","authors":"Octave Lacourte","doi":"10.4171/ggd/664","DOIUrl":"https://doi.org/10.4171/ggd/664","url":null,"abstract":"Let PC be the group of bijections from [0, 1[ to itself which are continuous outside a finite set. Let PC be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of PC vanishes. That is, the quotient map PC $rightarrow$ PC splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, from PC to Z 2Z. Then we use this signature to list normal subgroups of every subgroup G of PC which contains S fin and such that G, the projection of G in PC , is simple.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49489207","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":"No growth-gaps for special cube complexes","authors":"Jiakai Li, D. Wise","doi":"10.4171/ggd/537","DOIUrl":"https://doi.org/10.4171/ggd/537","url":null,"abstract":"","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/ggd/537","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44860923","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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