{"title":"Examples of geometric transition in low dimensions","authors":"Andrea Seppi","doi":"10.5802/tsg.368","DOIUrl":"https://doi.org/10.5802/tsg.368","url":null,"abstract":"The purpose of this note is to discuss examples of geometric transition from hyperbolic structures to half-pipe and Anti-de Sitter structures in dimensions two, three and four. As a warm-up, explicit examples of transition to Euclidean and spherical structures are presented. No new results appear here; nor an exhaustive treatment is aimed. On the other hand, details of some elementary computations are provided to explain certain techniques involved. This note, and in particular the last section, can also serve as an introduction to the ideas behind the four-dimensional construction of [RS19].","PeriodicalId":256700,"journal":{"name":"Séminaire de théorie spectrale et géométrie","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115631512","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}
{"title":"Existence of homogeneous metrics with prescribed Ricci curvature","authors":"M. Gould, A. Pulemotov","doi":"10.5802/TSG.313","DOIUrl":"https://doi.org/10.5802/TSG.313","url":null,"abstract":"— Consider a compact Lie group G and a closed subgroup H < G. Suppose T is a positive-definite G-invariant (0,2)-tensor field on the homogeneous space M = G/H. In this note, we state a sufficient condition for the existence of a G-invariant Riemannian metric on M whose Ricci curvature coincides with cT for some c > 0. This condition is, in fact, necessary if the isotropy representation of M splits into two inequivalent irreducible summands. After stating the main result, we work out an example.","PeriodicalId":256700,"journal":{"name":"Séminaire de théorie spectrale et géométrie","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127100367","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}
{"title":"A note on the relation between Hartnell’s firefighter problem and growth of groups","authors":"Eduardo Mart'inez-Pedroza","doi":"10.5802/TSG.314","DOIUrl":"https://doi.org/10.5802/TSG.314","url":null,"abstract":"The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell. The game on a graph $G$ can be described as follows: let $f_n$ be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time $ngeq 1$, $f_n$ vertices which are not on fire become protected, and then the fire spreads to all unprotected neighbors of vertices on fire; once a vertex is protected or is on fire, it remains so for all time intervals. The graph $G$ has the emph{$f_n$-containment property} if every initial fire admits an strategy that protects $f_n$ vertices at time $n$ so that the set of vertices on fire is eventually constant. If the graph $G$ has the containment property for a sequence of the form $f_n=Cn^d$, then the graph is said to have emph{polynomial containment}. In [5], it is shown that any locally finite graph with polynomial growth has polynomial containment; and it is remarked that the converse does not hold. That article also raised the question of whether the equivalence of polynomial growth and polynomial containment holds for Cayley graphs of finitely generated groups. In this short note, we remark how the equivalence holds for elementary amenable groups and for non-amenable groups from results in the literature.","PeriodicalId":256700,"journal":{"name":"Séminaire de théorie spectrale et géométrie","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115309010","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}
{"title":"Présentation","authors":"Pierre-Étienne Will, L. Rizzi","doi":"10.5802/tsg.359","DOIUrl":"https://doi.org/10.5802/tsg.359","url":null,"abstract":"","PeriodicalId":256700,"journal":{"name":"Séminaire de théorie spectrale et géométrie","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127681843","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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