We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups of finite-type surfaces, that is, those subgroups coming from the embeddings of closed subsurfaces. In the second half of the paper, we study under which hypotheses our amalgamation procedure preserves several notions of convexity in HHS, such as hierarchical quasiconvexity (as introduced by Behrstock, Hagen, and Sisto) and strong quasiconvexity (every quasigeodesic with endpoints on the subset lies in a uniform neighbourhood). This answers a question of Russell, Spriano, and Tran.
{"title":"A combination theorem for hierarchically quasiconvex subgroups, and application to geometric subgroups of mapping class groups","authors":"Giorgio Mangioni","doi":"arxiv-2409.03602","DOIUrl":"https://doi.org/arxiv-2409.03602","url":null,"abstract":"We provide sufficient conditions for two subgroups of a hierarchically\u0000hyperbolic group to generate an amalgamated free product over their\u0000intersection. The result applies in particular to certain geometric subgroups\u0000of mapping class groups of finite-type surfaces, that is, those subgroups\u0000coming from the embeddings of closed subsurfaces. In the second half of the paper, we study under which hypotheses our\u0000amalgamation procedure preserves several notions of convexity in HHS, such as\u0000hierarchical quasiconvexity (as introduced by Behrstock, Hagen, and Sisto) and\u0000strong quasiconvexity (every quasigeodesic with endpoints on the subset lies in\u0000a uniform neighbourhood). This answers a question of Russell, Spriano, and\u0000Tran.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206464","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 classify all finite groups such that all irreducible character degrees appear with multiplicity at most $2$. As a consequence, we prove that the largest group with at most $2$ irreducible characters of the same degree is the Baby Monster.
{"title":"The Baby Monster is the largest group with at most $2$ irreducible characters with the same degree","authors":"Juan Martínez Madrid","doi":"arxiv-2409.03345","DOIUrl":"https://doi.org/arxiv-2409.03345","url":null,"abstract":"We classify all finite groups such that all irreducible character degrees\u0000appear with multiplicity at most $2$. As a consequence, we prove that the\u0000largest group with at most $2$ irreducible characters of the same degree is the\u0000Baby Monster.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206467","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}
Tambara functors arise in equivariant homotopy theory as the structure adherent to the homotopy groups of a coherently commutative equivariant ring spectrum. We show that if $k$ is a field-like $C_{p^n}$-Tambara functor, then $k$ is the coinduction of a field-like $C_{p^s}$-Tambara functor $ell$ such that $ell(C_{p^s}/e)$ is a field. If this field has characteristic other than $p$, we observe that $ell$ must be a fixed-point Tambara functor, and if the characteristic is $p$, we determine all possible forms of $ell$ through an analysis of the behavior of the Frobenius endomorphism and an application of Artin-Schreier theory.
{"title":"A classification of $C_{p^n}$-Tambara fields","authors":"Noah Wisdom","doi":"arxiv-2409.02966","DOIUrl":"https://doi.org/arxiv-2409.02966","url":null,"abstract":"Tambara functors arise in equivariant homotopy theory as the structure\u0000adherent to the homotopy groups of a coherently commutative equivariant ring\u0000spectrum. We show that if $k$ is a field-like $C_{p^n}$-Tambara functor, then\u0000$k$ is the coinduction of a field-like $C_{p^s}$-Tambara functor $ell$ such\u0000that $ell(C_{p^s}/e)$ is a field. If this field has characteristic other than\u0000$p$, we observe that $ell$ must be a fixed-point Tambara functor, and if the\u0000characteristic is $p$, we determine all possible forms of $ell$ through an\u0000analysis of the behavior of the Frobenius endomorphism and an application of\u0000Artin-Schreier theory.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206469","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}
J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor's ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group $G$ on a spherically homogeneous rooted tree $T$ there is a canonical $G$-equivariant isomorphism between the Boolean algebra associated with the structure lattice of $G$ and the Boolean algebra of clopen subsets of the boundary of $T$.
{"title":"Branch actions and the structure lattice","authors":"Jorge Fariña-Asategui, Rostislav Grigorchuk","doi":"arxiv-2409.01655","DOIUrl":"https://doi.org/arxiv-2409.01655","url":null,"abstract":"J. S. Wilson proved in 1971 an isomorphism between the structural lattice\u0000associated to a group belonging to his second class of groups with every proper\u0000quotient finite and the Boolean algebra of clopen subsets of Cantor's ternary\u0000set. In this paper we generalize this isomorphism to the class of branch\u0000groups. Moreover, we show that for every faithful branch action of a group $G$\u0000on a spherically homogeneous rooted tree $T$ there is a canonical\u0000$G$-equivariant isomorphism between the Boolean algebra associated with the\u0000structure lattice of $G$ and the Boolean algebra of clopen subsets of the\u0000boundary of $T$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206471","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 study the double-coset zeta functions for groups acting on trees, focusing mainly on weakly locally $infty$-transitive or (P)-closed actions. After giving a geometric characterisation of convergence for the defining series, we provide explicit determinant formulae for the relevant zeta functions in terms of local data of the action. Moreover, we prove that evaluation at $-1$ satisfies the expected identity with the Euler-Poincar'e characteristic of the group. The behaviour at $-1$ also sheds light on a connection with the Ihara zeta function of a weighted graph introduced by A. Deitmar.
{"title":"Double-coset zeta functions for groups acting on trees","authors":"Bianca Marchionna","doi":"arxiv-2409.01860","DOIUrl":"https://doi.org/arxiv-2409.01860","url":null,"abstract":"We study the double-coset zeta functions for groups acting on trees, focusing\u0000mainly on weakly locally $infty$-transitive or (P)-closed actions. After\u0000giving a geometric characterisation of convergence for the defining series, we\u0000provide explicit determinant formulae for the relevant zeta functions in terms\u0000of local data of the action. Moreover, we prove that evaluation at $-1$\u0000satisfies the expected identity with the Euler-Poincar'e characteristic of the\u0000group. The behaviour at $-1$ also sheds light on a connection with the Ihara\u0000zeta function of a weighted graph introduced by A. Deitmar.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206470","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 provide a fine classification of rigid three-dimensional torus quotients with isolated canonical singularities, up to biholomorphism and diffeomorphism. This complements the classification of Calabi-Yau 3-folds of type $rm{III}_0$, which are those quotients with Gorenstein singularities.
{"title":"The Classification of Rigid Torus Quotients with Canonical Singularities in Dimension Three","authors":"Christian Gleissner, Julia Kotonski","doi":"arxiv-2409.01050","DOIUrl":"https://doi.org/arxiv-2409.01050","url":null,"abstract":"We provide a fine classification of rigid three-dimensional torus quotients\u0000with isolated canonical singularities, up to biholomorphism and diffeomorphism.\u0000This complements the classification of Calabi-Yau 3-folds of type $rm{III}_0$,\u0000which are those quotients with Gorenstein singularities.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206476","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}
Let $G leqslant {rm Sym}(Omega)$ be a finite transitive permutation group and recall that an element in $G$ is a derangement if it has no fixed points on $Omega$. Let $Delta(G)$ be the set of derangements in $G$ and define $delta(G) = |Delta(G)|/|G|$ and $Delta(G)^2 = { xy ,:, x,y in Delta(G)}$. In recent years, there has been a focus on studying derangements in simple groups, leading to several remarkable results. For example, by combining a theorem of Fulman and Guralnick with recent work by Larsen, Shalev and Tiep, it follows that $delta(G) geqslant 0.016$ and $G = Delta(G)^2$ for all sufficiently large simple transitive groups $G$. In this paper, we extend these results in several directions. For example, we prove that $delta(G) geqslant 89/325$ and $G = Delta(G)^2$ for all finite simple primitive groups with soluble point stabilisers, without any order assumptions, and we show that the given lower bound on $delta(G)$ is best possible. We also prove that every finite simple transitive group can be generated by two conjugate derangements, and we present several new results on derangements in arbitrary primitive permutation groups.
{"title":"On derangements in simple permutation groups","authors":"Timothy C. Burness, Marco Fusari","doi":"arxiv-2409.01043","DOIUrl":"https://doi.org/arxiv-2409.01043","url":null,"abstract":"Let $G leqslant {rm Sym}(Omega)$ be a finite transitive permutation group\u0000and recall that an element in $G$ is a derangement if it has no fixed points on\u0000$Omega$. Let $Delta(G)$ be the set of derangements in $G$ and define\u0000$delta(G) = |Delta(G)|/|G|$ and $Delta(G)^2 = { xy ,:, x,y in\u0000Delta(G)}$. In recent years, there has been a focus on studying derangements\u0000in simple groups, leading to several remarkable results. For example, by\u0000combining a theorem of Fulman and Guralnick with recent work by Larsen, Shalev\u0000and Tiep, it follows that $delta(G) geqslant 0.016$ and $G = Delta(G)^2$ for\u0000all sufficiently large simple transitive groups $G$. In this paper, we extend\u0000these results in several directions. For example, we prove that $delta(G)\u0000geqslant 89/325$ and $G = Delta(G)^2$ for all finite simple primitive groups\u0000with soluble point stabilisers, without any order assumptions, and we show that\u0000the given lower bound on $delta(G)$ is best possible. We also prove that every\u0000finite simple transitive group can be generated by two conjugate derangements,\u0000and we present several new results on derangements in arbitrary primitive\u0000permutation groups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206472","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 give characterizations of the notions of van der Corput (vdC) set, nice vdC set and set of nice recurrence (defined below) in countable amenable groups. This allows us to prove that nice vdC sets are sets of nice recurrence and that vdC sets are independent of the F{o}lner sequence used to define them, answering questions from Bergelson and Lesigne in the context of countable amenable groups. We also give a spectral characterization of vdC sets in abelian groups. The methods developed in this paper allow us to establish a converse to the Furstenberg correspondence principle. In addition, we introduce vdC sets in general non amenable groups and establish some basic properties of them, such as partition regularity. Several results in this paper, including the converse to Furstenberg's correspondence principle, have also been proved independently by Robin Tucker-Drob and Sohail Farhangi in their article `Van der Corput sets in amenable groups and beyond', which is being uploaded to arXiv simultaneously to this one.
{"title":"An inverse of Furstenberg's correspondence principle and applications to van der Corput sets","authors":"Saúl Rodríguez Martín","doi":"arxiv-2409.00885","DOIUrl":"https://doi.org/arxiv-2409.00885","url":null,"abstract":"In this article we give characterizations of the notions of van der Corput\u0000(vdC) set, nice vdC set and set of nice recurrence (defined below) in countable\u0000amenable groups. This allows us to prove that nice vdC sets are sets of nice\u0000recurrence and that vdC sets are independent of the F{o}lner sequence used to\u0000define them, answering questions from Bergelson and Lesigne in the context of\u0000countable amenable groups. We also give a spectral characterization of vdC sets\u0000in abelian groups. The methods developed in this paper allow us to establish a\u0000converse to the Furstenberg correspondence principle. In addition, we introduce\u0000vdC sets in general non amenable groups and establish some basic properties of\u0000them, such as partition regularity. Several results in this paper, including the converse to Furstenberg's\u0000correspondence principle, have also been proved independently by Robin\u0000Tucker-Drob and Sohail Farhangi in their article `Van der Corput sets in\u0000amenable groups and beyond', which is being uploaded to arXiv simultaneously to\u0000this one.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206473","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 characterize the normal extensions of inverse semigroups isomorphic to full restricted semidirect products, and present a Kalouznin-Krasner theorem which holds for a wider class of normal extensions of inverse semigroups than that in the well-known embedding theorem due to Billhardt, and also strengthens that result in two respects. First, the wreath product construction applied in our result, and stemmming from Houghton's wreath product, is a full restricted semidirect product not merely a lambda-semidirect product. Second, the Kernel classes of our wreath product construction are direct products of some Kernel classes of the normal extension to be embedded rather than only inverse subsemigroups of the direct power of its whole Kernel.
{"title":"Normal extensions and full restricted semidirect products of inverse semigroups","authors":"Mária B. Szendrei","doi":"arxiv-2409.00870","DOIUrl":"https://doi.org/arxiv-2409.00870","url":null,"abstract":"We characterize the normal extensions of inverse semigroups isomorphic to\u0000full restricted semidirect products, and present a Kalouznin-Krasner theorem\u0000which holds for a wider class of normal extensions of inverse semigroups than\u0000that in the well-known embedding theorem due to Billhardt, and also strengthens\u0000that result in two respects. First, the wreath product construction applied in\u0000our result, and stemmming from Houghton's wreath product, is a full restricted\u0000semidirect product not merely a lambda-semidirect product. Second, the Kernel\u0000classes of our wreath product construction are direct products of some Kernel\u0000classes of the normal extension to be embedded rather than only inverse\u0000subsemigroups of the direct power of its whole Kernel.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"406 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206474","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 answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the Fo lner sequence used to define it. This result has been discovered independently by Sa'ul Rodr'iguez Mart'in. Both ours and Rodr'iguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in $mathbb{N}$ that do not involve Fo lner sequences remain equivalent for arbitrary countably infinite groups.
{"title":"Asymptotic dynamics on amenable groups and van der Corput sets","authors":"Sohail Farhangi, Robin Tucker-Drob","doi":"arxiv-2409.00806","DOIUrl":"https://doi.org/arxiv-2409.00806","url":null,"abstract":"We answer a question of Bergelson and Lesigne by showing that the notion of\u0000van der Corput set does not depend on the Fo lner sequence used to define it.\u0000This result has been discovered independently by Sa'ul Rodr'iguez Mart'in.\u0000Both ours and Rodr'iguez's proofs proceed by first establishing a converse to\u0000the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent\u0000sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der\u0000Corput sets in $mathbb{N}$ that do not involve Fo lner sequences remain\u0000equivalent for arbitrary countably infinite groups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206477","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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