What's Next?Pub Date : 2015-05-28DOI: 10.1515/9780691185897-003
M. Boileau, Stefan Friedl
{"title":"The Profinite Completion of 3-Manifold Groups, Fiberedness and the Thurston Norm","authors":"M. Boileau, Stefan Friedl","doi":"10.1515/9780691185897-003","DOIUrl":"https://doi.org/10.1515/9780691185897-003","url":null,"abstract":"We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133096041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
What's Next?Pub Date : 2015-03-18DOI: 10.1515/9780691185897-009
R. Kenyon
{"title":"Right-Angled Hexagon Tilings of the Hyperbolic Plane","authors":"R. Kenyon","doi":"10.1515/9780691185897-009","DOIUrl":"https://doi.org/10.1515/9780691185897-009","url":null,"abstract":"We study isometry-invariant probability measures on the space $Omega$ of tilings of the hyperbolic plane with right-angled hexagons of varying shapes. We prove that, for each measure $mu$ in a certain natural family of measures on right-angled hexagons, there is an isometry-invariant measure on $Omega$ whose marginal distribution on tiles is $mu$.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128331053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
What's Next?Pub Date : 2015-02-23DOI: 10.2307/j.ctvthhdvv.8
V. Delecroix, A. Zorich
{"title":"Cries and Whispers in Wind-Tree Forests","authors":"V. Delecroix, A. Zorich","doi":"10.2307/j.ctvthhdvv.8","DOIUrl":"https://doi.org/10.2307/j.ctvthhdvv.8","url":null,"abstract":"We study billiard in the plane endowed with symmetric $mathbb{Z}^2$-periodic obstacles of a right-angled polygonal shape. One of our main interests is the dependence of the diffusion rate of the billiard on the shape of the obstacle. We prove, in particular, that when the number of angles of a symmetric connected obstacle grows, the diffusion rate tends to zero, thus answering a question of J.-C. Yoccoz. \u0000Our results are based on computation of Lyapunov exponents of the Hodge bundle over hyperelliptic loci in the moduli spaces of quadratic differentials, which represents independent interest. In particular, we compute the exact value of the Lyapunov exponent $lambda^+_1$ for all elliptic loci of quadratic differentials with simple zeroes and poles.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130788847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
What's Next?Pub Date : 2015-02-17DOI: 10.2307/j.ctvthhdvv.12
Sarah C. Koch, Tan Lei
{"title":"On Balanced Planar Graphs, Following W. Thurston","authors":"Sarah C. Koch, Tan Lei","doi":"10.2307/j.ctvthhdvv.12","DOIUrl":"https://doi.org/10.2307/j.ctvthhdvv.12","url":null,"abstract":"Let f : S 2 → S 2 be an orientation-preserving branched covering map of degree d ≥ 2, and let Σ be an oriented Jordan curve passing through the critical values of f . Then Γ := f −1 (Σ) is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such Γ, where f has 2d−2 distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124755877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
What's Next?Pub Date : 2015-01-09DOI: 10.1515/9780691185897-002
I. Agol, N. Dunfield
{"title":"Certifying the Thurston Norm via SL(2,ℂ)-twisted Homology","authors":"I. Agol, N. Dunfield","doi":"10.1515/9780691185897-002","DOIUrl":"https://doi.org/10.1515/9780691185897-002","url":null,"abstract":"We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large class of hyperbolic knots in the 3-sphere which includes all special arborescent knots and many knots whose ordinary Alexander polynomial is trivial. This theorem follows from results showing that the tautness of certain sutured manifolds can be certified by checking that they are a product from the point of view of homology with coefficients twisted by an SL(2, C)-representation.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123177303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
What's Next?Pub Date : 2014-11-06DOI: 10.1515/9780691185897-005
Danny Calegari
{"title":"Coxeter Groups and Random Groups","authors":"Danny Calegari","doi":"10.1515/9780691185897-005","DOIUrl":"https://doi.org/10.1515/9780691185897-005","url":null,"abstract":"For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any density less than a half (or in the few relators model) contains quasiconvex subgroups commensurable with some member of the family, with overwhelming probability.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"438 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116328998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
What's Next?Pub Date : 2014-07-02DOI: 10.2307/j.ctvthhdvv.14
Yi Ni, Xingru Zhang
{"title":"Dehn Surgery on Knots in S³ Producing Nil Seifert Fibered Spaces","authors":"Yi Ni, Xingru Zhang","doi":"10.2307/j.ctvthhdvv.14","DOIUrl":"https://doi.org/10.2307/j.ctvthhdvv.14","url":null,"abstract":"We prove that there are exactly 6 Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in S, with {60, 144, 156, 288, 300} as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly 4 specific hyperbolic knots in S which admit Nil Seifert fibred surgery. We also give some more general results and a more general conjecture concerning Seifert fibred surgeries on hyperbolic knots in S.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"213 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115276318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
What's Next?Pub Date : 2001-07-11DOI: 10.1515/9780691185897-007
Benson Farb, J. Franks
{"title":"Groups of Homeomorphisms of One-Manifolds, I: Actions of Nonlinear Groups","authors":"Benson Farb, J. Franks","doi":"10.1515/9780691185897-007","DOIUrl":"https://doi.org/10.1515/9780691185897-007","url":null,"abstract":"This self-contained paper is part of a series cite{FF2,FF3} on actions by diffeomorphisms of infinite groups on compact manifolds. The two main results presented here are: \u00001) Any homomorphism of (almost any) mapping class group or automorphism group of a free group into $Diff_+^r(S^1), rgeq 2$ is trivial. For r=0 Nielsen showed that in many cases nontrivial (even faithful) representations exist. Somewhat weaker results are proven for finite index subgroups. \u00002) We construct a finitely-presented group of real-analytic diffeomorphisms of $R$ which is not residually finite.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121690331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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