In this paper we go over the history of the Fuglede or Spectral Set Conjecture as it has developed over the last 30 years or so. We do not aim to be exhaustive and we do not cover important areas of development such as the results on the problem in classes of finite groups or the version of the problem that focuses on spectral measures instead of sets. The selection of the material has been strongly influenced by personal taste, history and capabilities. We are trying to be more descriptive than detailed and we point out several open questions.
{"title":"Orthogonal Fourier Analysis on Domains","authors":"Mihail N. Kolountzakis","doi":"arxiv-2408.15361","DOIUrl":"https://doi.org/arxiv-2408.15361","url":null,"abstract":"In this paper we go over the history of the Fuglede or Spectral Set\u0000Conjecture as it has developed over the last 30 years or so. We do not aim to\u0000be exhaustive and we do not cover important areas of development such as the\u0000results on the problem in classes of finite groups or the version of the\u0000problem that focuses on spectral measures instead of sets. The selection of the\u0000material has been strongly influenced by personal taste, history and\u0000capabilities. We are trying to be more descriptive than detailed and we point\u0000out several open questions.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187635","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}
We define the chain Sobolev space on a possibly non-complete metric measure space in terms of chain upper gradients. In this context, $varepsilon$-chains are a finite collection of points with distance at most $varepsilon$ between consecutive points. They play the role of discrete versions of curves. Chain upper gradients are defined accordingly and the chain Sobolev space is defined by letting the size parameter $varepsilon$ going to zero. In the complete setting, we prove that the chain Sobolev space is equal to the classical notions of Sobolev spaces in terms of relaxation of upper gradients or of the local Lipschitz constant of Lipschitz functions. The proof of this fact is inspired by a recent technique developed by Eriksson-Bique. In the possible non-complete setting, we prove that the chain Sobolev space is equal to the one defined via relaxation of the local Lipschitz constant of Lipschitz functions, while in general they are different from the one defined via upper gradients along curves. We apply the theory developed in the paper to prove equivalent formulations of the Poincar'{e} inequality in terms of pointwise estimates involving $varepsilon$-upper gradients, lower bounds on modulus of chains connecting points and size of separating sets measured with the Minkowski content in the non-complete setting. Along the way, we discuss the notion of weak $varepsilon$-upper gradients and asymmetric notions of integral along chains.
{"title":"Sobolev spaces via chains in metric measure spaces","authors":"Emanuele Caputo, Nicola Cavallucci","doi":"arxiv-2408.15071","DOIUrl":"https://doi.org/arxiv-2408.15071","url":null,"abstract":"We define the chain Sobolev space on a possibly non-complete metric measure\u0000space in terms of chain upper gradients. In this context, $varepsilon$-chains\u0000are a finite collection of points with distance at most $varepsilon$ between\u0000consecutive points. They play the role of discrete versions of curves. Chain\u0000upper gradients are defined accordingly and the chain Sobolev space is defined\u0000by letting the size parameter $varepsilon$ going to zero. In the complete\u0000setting, we prove that the chain Sobolev space is equal to the classical\u0000notions of Sobolev spaces in terms of relaxation of upper gradients or of the\u0000local Lipschitz constant of Lipschitz functions. The proof of this fact is\u0000inspired by a recent technique developed by Eriksson-Bique. In the possible\u0000non-complete setting, we prove that the chain Sobolev space is equal to the one\u0000defined via relaxation of the local Lipschitz constant of Lipschitz functions,\u0000while in general they are different from the one defined via upper gradients\u0000along curves. We apply the theory developed in the paper to prove equivalent\u0000formulations of the Poincar'{e} inequality in terms of pointwise estimates\u0000involving $varepsilon$-upper gradients, lower bounds on modulus of chains\u0000connecting points and size of separating sets measured with the Minkowski\u0000content in the non-complete setting. Along the way, we discuss the notion of\u0000weak $varepsilon$-upper gradients and asymmetric notions of integral along\u0000chains.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187639","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}
Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Paul Wollan
We prove that there is a function $f$ such that every graph with no $K$-fat $K_4$ minor is $f(K)$-quasi-isometric to a graph with no $K_4$ minor. This solves the $K_4$-case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective $K_4^-$-case, which was first established by Fujiwara and Papasoglu.
{"title":"A characterisation of graphs quasi-isometric to $K_4$-minor-free graphs","authors":"Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Paul Wollan","doi":"arxiv-2408.15335","DOIUrl":"https://doi.org/arxiv-2408.15335","url":null,"abstract":"We prove that there is a function $f$ such that every graph with no $K$-fat\u0000$K_4$ minor is $f(K)$-quasi-isometric to a graph with no $K_4$ minor. This\u0000solves the $K_4$-case of a general conjecture of Georgakopoulos and Papasoglu.\u0000Our proof technique also yields a new short proof of the respective\u0000$K_4^-$-case, which was first established by Fujiwara and Papasoglu.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"313 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187641","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}
Discrete geometries in hyperbolic space are of longstanding interest in pure mathematics and have come to recent attention in holography, quantum information, and condensed matter physics. Working at a purely geometric level, we describe how any regular tessellation of ($d+1$)-dimensional hyperbolic space naturally admits a $d$-dimensional boundary geometry with self-similar ''quasicrystalline'' properties. In particular, the boundary geometry is described by a local, invertible, self-similar substitution tiling, that discretizes conformal geometry. We greatly refine an earlier description of these local substitution rules that appear in the 1D/2D example and use the refinement to give the first extension to higher dimensional bulks; including a detailed account for all regular 3D hyperbolic tessellations. We comment on global issues, including the reconstruction of bulk geometries from boundary data, and introduce the notion of a ''holographic foliation'': a foliation by a stack of self-similar quasicrystals, where the full geometry of the bulk (and of the foliation itself) is encoded in any single leaf in a local invertible way. In the ${3,5,3}$ tessellation of 3D hyperbolic space by regular icosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold symmetry which is not the Penrose tiling, and record and comment on a related conjecture of William Thurston. We end with a large list of open questions for future analytic and numerical studies.
{"title":"Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs","authors":"Latham Boyle, Justin Kulp","doi":"arxiv-2408.15316","DOIUrl":"https://doi.org/arxiv-2408.15316","url":null,"abstract":"Discrete geometries in hyperbolic space are of longstanding interest in pure\u0000mathematics and have come to recent attention in holography, quantum\u0000information, and condensed matter physics. Working at a purely geometric level,\u0000we describe how any regular tessellation of ($d+1$)-dimensional hyperbolic\u0000space naturally admits a $d$-dimensional boundary geometry with self-similar\u0000''quasicrystalline'' properties. In particular, the boundary geometry is\u0000described by a local, invertible, self-similar substitution tiling, that\u0000discretizes conformal geometry. We greatly refine an earlier description of\u0000these local substitution rules that appear in the 1D/2D example and use the\u0000refinement to give the first extension to higher dimensional bulks; including a\u0000detailed account for all regular 3D hyperbolic tessellations. We comment on\u0000global issues, including the reconstruction of bulk geometries from boundary\u0000data, and introduce the notion of a ''holographic foliation'': a foliation by a\u0000stack of self-similar quasicrystals, where the full geometry of the bulk (and\u0000of the foliation itself) is encoded in any single leaf in a local invertible\u0000way. In the ${3,5,3}$ tessellation of 3D hyperbolic space by regular\u0000icosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold\u0000symmetry which is not the Penrose tiling, and record and comment on a related\u0000conjecture of William Thurston. We end with a large list of open questions for\u0000future analytic and numerical studies.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187634","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}
The class of uniformly smooth hyperbolic spaces was recently introduced by Pinto as a common generalization of both CAT(0) spaces and uniformly smooth Banach spaces, in a way that Reich's theorem on resolvent convergence could still be proven. We define products of such spaces, showing that they are reasonably well-behaved. In this manner, we provide the first example of a space for which Reich's theorem holds and which is neither a CAT(0) space, nor a convex subset of a normed space.
{"title":"Products of hyperbolic spaces","authors":"Andrei Sipos","doi":"arxiv-2408.14093","DOIUrl":"https://doi.org/arxiv-2408.14093","url":null,"abstract":"The class of uniformly smooth hyperbolic spaces was recently introduced by\u0000Pinto as a common generalization of both CAT(0) spaces and uniformly smooth\u0000Banach spaces, in a way that Reich's theorem on resolvent convergence could\u0000still be proven. We define products of such spaces, showing that they are\u0000reasonably well-behaved. In this manner, we provide the first example of a\u0000space for which Reich's theorem holds and which is neither a CAT(0) space, nor\u0000a convex subset of a normed space.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187636","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}
In this short note we survey theorems and provide conjectures on gluing constructions under lower curvature bounds in smooth and non-smooth context. Focusing on synthetic lower Ricci curvature bounds we consider Riemannian manifolds, weighted Riemannian manifolds, Alexandrov spaces, collapsed and non-collapsed $RCD$ spaces, and sub-Riemannian spaces.
{"title":"Glued spaces and lower curvature bounds","authors":"Christian Ketterer","doi":"arxiv-2408.13137","DOIUrl":"https://doi.org/arxiv-2408.13137","url":null,"abstract":"In this short note we survey theorems and provide conjectures on gluing\u0000constructions under lower curvature bounds in smooth and non-smooth context.\u0000Focusing on synthetic lower Ricci curvature bounds we consider Riemannian\u0000manifolds, weighted Riemannian manifolds, Alexandrov spaces, collapsed and\u0000non-collapsed $RCD$ spaces, and sub-Riemannian spaces.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"2022 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187640","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}
In this paper we present a unique 4-dimensional body of constant width based on the classical notion of focal conics.
在本文中，我们根据经典的焦点圆锥概念，提出了一个独特的恒宽四维体。
{"title":"A 4-Dimensional Peabody of Constant Width","authors":"Isaac Arelio, Luis Montejano, Deborah Oliveros","doi":"arxiv-2408.13241","DOIUrl":"https://doi.org/arxiv-2408.13241","url":null,"abstract":"In this paper we present a unique 4-dimensional body of constant width based\u0000on the classical notion of focal conics.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187637","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}
We introduce the notion of "Banach metrics" on finitely generated infinite groups. This extends the notion of a Cayley graph (as a metric space). Our motivation comes from trying to detect the existence of virtual homomorphisms into Z, the additive group of integers. We show that detection of such homomorphisms through metric functional boundaries of Cayley graphs isn't always possible. However, we prove that it is always possible to do so through a metric functional boundary of some Banach metric on the group.
{"title":"Detecting virtual homomorphisms via Banach metrics","authors":"Liran Ron-George, Ariel Yadin","doi":"arxiv-2408.11543","DOIUrl":"https://doi.org/arxiv-2408.11543","url":null,"abstract":"We introduce the notion of \"Banach metrics\" on finitely generated infinite\u0000groups. This extends the notion of a Cayley graph (as a metric space). Our\u0000motivation comes from trying to detect the existence of virtual homomorphisms\u0000into Z, the additive group of integers. We show that detection of such\u0000homomorphisms through metric functional boundaries of Cayley graphs isn't\u0000always possible. However, we prove that it is always possible to do so through\u0000a metric functional boundary of some Banach metric on the group.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187642","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}
In this article, we found all simple closed geodesics on regular spherical octahedra and spherical cubes. In addition, we estimate the number of simple closed geodesics on regular spherical tetrahedra.
{"title":"Simple closed geodesics on regular spherical polyhedra","authors":"Darya Sukhorebska","doi":"arxiv-2408.10782","DOIUrl":"https://doi.org/arxiv-2408.10782","url":null,"abstract":"In this article, we found all simple closed geodesics on regular spherical\u0000octahedra and spherical cubes. In addition, we estimate the number of simple\u0000closed geodesics on regular spherical tetrahedra.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187644","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}
We show that a finitely presented group virtually admits a planar Cayley graph if and only if it is asymptotically minor-excluded, partially answering a conjecture of Georgakopoulos and Papasoglu in the affirmative.
{"title":"Fat minors in finitely presented groups","authors":"Joseph MacManus","doi":"arxiv-2408.10748","DOIUrl":"https://doi.org/arxiv-2408.10748","url":null,"abstract":"We show that a finitely presented group virtually admits a planar Cayley\u0000graph if and only if it is asymptotically minor-excluded, partially answering a\u0000conjecture of Georgakopoulos and Papasoglu in the affirmative.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187651","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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