{"title":"On topological full groups of $mathbb Z^d$-actions","authors":"M. Chornyi, K. Juschenko, V. Nekrashevych","doi":"10.4171/ggd/534","DOIUrl":"https://doi.org/10.4171/ggd/534","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/534","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43915131","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}
Francesco Matucci, Altair Santos de Oliveira-Tosti
Monod introduced in [14] a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and invariant and use them to compute centralizers in one group of this family.
{"title":"Conjugacy and centralizers in groups of piecewise projective homeomorphisms","authors":"Francesco Matucci, Altair Santos de Oliveira-Tosti","doi":"10.4171/ggd/657","DOIUrl":"https://doi.org/10.4171/ggd/657","url":null,"abstract":"Monod introduced in [14] a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and invariant and use them to compute centralizers in one group of this family.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48841154","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 note we present new asymptotic estimates comparing the word length and geodesic length of closed geodesics on surfaces with (variable) negative sectional curvatures. In particular, we provide an averaged comparison of these two important quantities and obtain precise statistical results, including a central limit theorem and a local limit theorem. Further, as a corollary we also improve an asymptotic formula of R. Sharp and the second author. Finally, we relate our results to recent work of Gekhtman, Taylor and Tiozzo.
{"title":"Comparison theorems for closed geodesics on negatively curved surfaces","authors":"S. Cantrell, M. Pollicott","doi":"10.4171/ggd/671","DOIUrl":"https://doi.org/10.4171/ggd/671","url":null,"abstract":"In this note we present new asymptotic estimates comparing the word length and geodesic length of closed geodesics on surfaces with (variable) negative sectional curvatures. In particular, we provide an averaged comparison of these two important quantities and obtain precise statistical results, including a central limit theorem and a local limit theorem. Further, as a corollary we also improve an asymptotic formula of R. Sharp and the second author. Finally, we relate our results to recent work of Gekhtman, Taylor and Tiozzo.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41983272","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 k be a number field, G an algebraic group defined over k, and G(k) the group of k-rational points in G. We determine the set of functions on G(k) which are of positive type and conjugation invariant, under the assumption that G(k) is generated by its unipotent elements. An essential step in the proof is the classification of the G(k)-invariant ergodic probability measures on an adelic solenoid naturally associated to G(k); this last result is deduced from Ratner's measure rigidity theorem for homogeneous spaces of S-adic Lie groups.
{"title":"Characters of algebraic groups over number fields","authors":"Bachir Bekka, Camille Francini","doi":"10.4171/ggd/678","DOIUrl":"https://doi.org/10.4171/ggd/678","url":null,"abstract":"Let k be a number field, G an algebraic group defined over k, and G(k) the group of k-rational points in G. We determine the set of functions on G(k) which are of positive type and conjugation invariant, under the assumption that G(k) is generated by its unipotent elements. An essential step in the proof is the classification of the G(k)-invariant ergodic probability measures on an adelic solenoid naturally associated to G(k); this last result is deduced from Ratner's measure rigidity theorem for homogeneous spaces of S-adic Lie groups.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48169315","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 a description of the automorphism group of a Rauzy diagram as a subgroup of the symmetric group. This is based on an example that appear in some personnal notes of Yoccoz that are to be published in the project ''Yoccoz archives''.
{"title":"The automorphism group of Rauzy diagrams","authors":"Corentin Boissy","doi":"10.4171/ggd/728","DOIUrl":"https://doi.org/10.4171/ggd/728","url":null,"abstract":"We give a description of the automorphism group of a Rauzy diagram as a subgroup of the symmetric group. This is based on an example that appear in some personnal notes of Yoccoz that are to be published in the project ''Yoccoz archives''.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47133199","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 families of finitely presented groups exhibiting new divergence behavior; we obtain divergence functions of the form $r^alpha$ for a dense set of exponents $alpha in [2,infty)$ and $r^nlog(r)$ for integers $n geq 2$. The same construction also yields examples of finitely presented groups which contain Morse elements that are not contracting.
{"title":"Divergence of finitely presented groups","authors":"N. Brady, H. Tran","doi":"10.4171/ggd/632","DOIUrl":"https://doi.org/10.4171/ggd/632","url":null,"abstract":"We construct families of finitely presented groups exhibiting new divergence behavior; we obtain divergence functions of the form $r^alpha$ for a dense set of exponents $alpha in [2,infty)$ and $r^nlog(r)$ for integers $n geq 2$. The same construction also yields examples of finitely presented groups which contain Morse elements that are not contracting.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44106456","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}
A non-trivial finitely generated pro-$p$ group $G$ is said to be strongly hereditarily self-similar of index $p$ if every non-trivial finitely generated closed subgroup of $G$ admits a faithful self-similar action on a $p$-ary tree. We classify the solvable torsion-free $p$-adic analytic pro-$p$ groups of dimension less than $p$ that are strongly hereditarily self-similar of index $p$. Moreover, we show that a solvable torsion-free $p$-adic analytic pro-$p$ group of dimension less than $p$ is strongly hereditarily self-similar of index $p$ if and only if it is isomorphic to the maximal pro-$p$ Galois group of some field that contains a primitive $p$-th root of unity. As a key step for the proof of the above results, we classify the 3-dimensional solvable torsion-free $p$-adic analytic pro-$p$ groups that admit a faithful self-similar action on a $p$-ary tree, completing the classification of the 3-dimensional torsion-free $p$-adic analytic pro-$p$ groups that admit such actions.
{"title":"On hereditarily self-similar $p$-adic analytic pro-$p$ groups","authors":"Francesco Noseda, I. Snopce","doi":"10.4171/ggd/641","DOIUrl":"https://doi.org/10.4171/ggd/641","url":null,"abstract":"A non-trivial finitely generated pro-$p$ group $G$ is said to be strongly hereditarily self-similar of index $p$ if every non-trivial finitely generated closed subgroup of $G$ admits a faithful self-similar action on a $p$-ary tree. We classify the solvable torsion-free $p$-adic analytic pro-$p$ groups of dimension less than $p$ that are strongly hereditarily self-similar of index $p$. Moreover, we show that a solvable torsion-free $p$-adic analytic pro-$p$ group of dimension less than $p$ is strongly hereditarily self-similar of index $p$ if and only if it is isomorphic to the maximal pro-$p$ Galois group of some field that contains a primitive $p$-th root of unity. As a key step for the proof of the above results, we classify the 3-dimensional solvable torsion-free $p$-adic analytic pro-$p$ groups that admit a faithful self-similar action on a $p$-ary tree, completing the classification of the 3-dimensional torsion-free $p$-adic analytic pro-$p$ groups that admit such actions.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44249529","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 cocycles of countable groups $Gamma$ of Borel automorphisms of a standard Borel space $(X, mathcal{B})$ taking values in a locally compact second countable group $G$. We prove that for a hyperfinite group $Gamma$ the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the $2$-odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup $H$ of $G$. We also provide a Borel version of Gottschalk-Hedlund theorem.
{"title":"Cohomology of hyperfinite Borel actions","authors":"S. Bezuglyi, S. Sanadhya","doi":"10.4171/ggd/633","DOIUrl":"https://doi.org/10.4171/ggd/633","url":null,"abstract":"We study cocycles of countable groups $Gamma$ of Borel automorphisms of a standard Borel space $(X, mathcal{B})$ taking values in a locally compact second countable group $G$. We prove that for a hyperfinite group $Gamma$ the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the $2$-odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup $H$ of $G$. We also provide a Borel version of Gottschalk-Hedlund theorem.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47753444","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 uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $mathbb{R}$-covered and we give a new description of the universal circle of $mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$.
{"title":"Minimality of the action on the universal circle of uniform foliations","authors":"Sérgio R. Fenley, R. Potrie","doi":"10.4171/ggd/637","DOIUrl":"https://doi.org/10.4171/ggd/637","url":null,"abstract":"Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $mathbb{R}$-covered and we give a new description of the universal circle of $mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46892452","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 several results of Helfer, Wise, Louder and Wilton related to coherence in one-relator groups to the more general setting of one-relator products of locally indicable groups. The methods developed to do so also give rise to a new proof of a theorem of Brodsky.
{"title":"Coherence and one-relator products of locally indicable groups","authors":"J. Howie, H. Short","doi":"10.4171/ggd/725","DOIUrl":"https://doi.org/10.4171/ggd/725","url":null,"abstract":"We extend several results of Helfer, Wise, Louder and Wilton related to coherence in one-relator groups to the more general setting of one-relator products of locally indicable groups. The methods developed to do so also give rise to a new proof of a theorem of Brodsky.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43343792","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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