Survey on Geometric Group Theory
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Arxiv:1704.05304V2 [Math.GR] 24 Aug 2017 Iergop,Rltvl Yeblcgroups
LINEAR GROUPS, CONJUGACY GROWTH, AND CLASSIFYING SPACES FOR FAMILIES OF SUBGROUPS TIMM VON PUTTKAMER AND XIAOLEI WU Abstract. Given a group G and a family of subgroups F , we consider its classifying space EF G with respect to F . When F = VCyc is the family of virtually cyclic subgroups, Juan- Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by L¨uck–Reich–Rognes–Varisco for the family of cyclic subgroups. Finally, we construct finitely generated groups that exhibit wild inner automorphims but which admit a model for EVCyc(G) whose 0-skeleton is finite. Introduction Given a group G, a family F of subgroups of G is a set of subgroups of G which is closed under conjugation and taking subgroups. We denote by EF (G)a G-CW-model for the classifying space for the family F . The space EF (G) is characterized by the property H that the fixed point set EF (G) is contractible for any H ∈F and empty otherwise. Recall that a G-CW-complex X is said to be finite if it has finitely many orbits of cells. Similarly, X is said to be of finite type if it has finitely many orbits of cells of dimension n for any n. We abbreviate EF (G) by EG for F = VCyc the family of virtually cyclic subgroups, EG for F = F in the family of finite subgroups and EG for F the family consisting only of the trivial subgroup. -
Serre's Conjecture II: a Survey
Serre’s conjecture II: a survey Philippe Gille Dedicated to Parimala Abstract We provide a survey of Serre’s conjecture II (1962) on the vanishing of Galois cohomology for simply connected semisimple groups defined over a field of cohomological dimension 2. ≤ Math subject classification: 20G05. 1 Introduction Serre’s original conjecture II (1962) states that the Galois cohomology set H1(k,G) vanishes for a semisimple simply connected algebraic group G defined over a perfect field of cohomological dimension 2 [53, 4.1] [54, II.3.1]. This means that G- ≤ § torsors (or principal homogeneous spaces) over Spec(k) are trivial. For example, if A is a central simple algebra defined over a field k and c k , ∈ × the subvariety X := nrd(y)=c GL (A) c { }⊂ 1 of elements of reduced norm c is a torsor under the special linear group G = SL1(A) which is semisimple and simply connected. If cd(k) 2, we thus expect that this ≤ G-torsor is trivial, i.e., X (k) = /0.Applying this to all c k , we thus expect that c ∈ × the reduced norm map A k is surjective. × → × For imaginary number fields, the surjectivity of reduced norms goes back to Eich- ler in 1938 (see [40, 5.4]). For function fields of complex surfaces, this follows § from the Tsen-Lang theorem because the reduced norm is a homogeneous form of degree deg(A) in deg(A)2-indeterminates [54, II.4.5]. In the general case, the surjec- Philippe Gille DMA, UMR 8553 du CNRS, Ecole normale superieure,´ 45 rue d’Ulm, F-75005 Paris, France e-mail: [email protected] 37 38 Philippe Gille tivity of reduced norms is due to Merkurjev-Suslin in 1981 [60, Th. -
Amenable Actions, Free Products and a Fixed Point Property
Submitted exclusively to the London Mathematical Society doi:10.1112/S0000000000000000 AMENABLE ACTIONS, FREE PRODUCTS AND A FIXED POINT PROPERTY Y. GLASNER and N. MONOD Abstract We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit interesting actions of that kind. A complete characterization of such free products is given in terms of a fixed point property. 1. Introduction 1.A. In the early 20th century, the construction of Lebesgue’s measure was followed by the discovery of the Banach-Hausdorff-Tarski paradoxes ([20], [16], [4], [30]; see also [18]). This prompted von Neumann [32] to study the following general question: Given a group G acting on a set X, when is there an invariant mean on X ? Definition 1.1. An invariant mean is a G-invariant map µ from the collection of subsets of X to [0, 1] such that (i) µ(A ∪ B) = µ(A) + µ(B) when A ∩ B = ∅ and (ii) µ(X) = 1. If such a mean exists, the action is called amenable. Remarks 1.2. (1) For the study of the classical paradoxes, one also considers normalisations other than (ii). (2) The notion of amenability later introduced by Zimmer [33] and its variants [3] are different, being in a sense dual to the above. 1.B. The thrust of von Neumann’s article was to show that the paradoxes, or lack thereof, originate in the structure of the group rather than the set X. He therefore proposed the study of amenable groups (then “meßbare Gruppen”), i.e. -
Cohomological Dimension of the Braid and Mapping Class Groups of Surfaces Daciberg Lima Gonçalves, John Guaschi, Miguel Maldonado
Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces Daciberg Lima Gonçalves, John Guaschi, Miguel Maldonado To cite this version: Daciberg Lima Gonçalves, John Guaschi, Miguel Maldonado. Embeddings and the (virtual) cohomo- logical dimension of the braid and mapping class groups of surfaces. Confluentes Mathematici, Institut Camille Jordan et Unité de Mathématiques Pures et Appliquées, 2018, 10 (1), pp.41-61. hal-01377681 HAL Id: hal-01377681 https://hal.archives-ouvertes.fr/hal-01377681 Submitted on 11 Oct 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EMBEDDINGS AND THE (VIRTUAL) COHOMOLOGICAL DIMENSION OF THE BRAID AND MAPPING CLASS GROUPS OF SURFACES DACIBERG LIMA GONC¸ALVES, JOHN GUASCHI, AND MIGUEL MALDONADO Abstract. In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of [4, 14] to show that the mapping class group MCG(N; k) of N relative to a k-point subset embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point subset. -
New York Journal of Mathematics Periodic Points on Shifts of Finite Type and Commensurability Invariants of Groups
New York Journal of Mathematics New York J. Math. 21 (2015) 811{822. Periodic points on shifts of finite type and commensurability invariants of groups David Carroll and Andrew Penland Abstract. We explore the relationship between subgroups and the pos- sible shifts of finite type (SFTs) which can be defined on a group. In particular, we investigate two group invariants, weak periodicity and strong periodicity, defined via symbolic dynamics on the group. We show that these properties are invariants of commensurability. Thus, many known results about periodic points in SFTs defined over groups are actually results about entire commensurability classes. Additionally, we show that the property of being not strongly periodic (i.e., the prop- erty of having a weakly aperiodic SFT) is preserved under extensions with finitely generated kernels. We conclude by raising questions and conjectures about the relationship of these invariants to the geometric notions of quasi-isometry and growth. Contents 1. Introduction 811 2. Background 814 2.1. Symbolic dynamics on groups 814 2.2. Quasi-isometry and commensurability of groups 815 3. Some special SFTs on groups 816 3.1. Higher block shift 816 3.2. Products of shifts 817 3.3. Locked shift 818 4. Commensurability and periodicity 818 4.1. SFTs and quotient groups 820 References 821 1. Introduction Given a group G and a finite alphabet A, a shift (over G) is a nonempty compact, G-equivariant subset of the full shift AG, where AG is a topological space regarded as a product of discrete spaces. A shift of finite type (SFT) Received April 6, 2015. -
Topics for the NTCIR-10 Math Task Full-Text Search Queries
Topics for the NTCIR-10 Math Task Full-Text Search Queries Michael Kohlhase (Editor) Jacobs University http://kwarc.info/kohlhase December 6, 2012 Abstract This document presents the challenge queries for the Math Information Retrieval Subtask in the NTCIR Math Task. Chapter 1 Introduction This document presents the challenge queries for the Math Information Retrieval (MIR) Subtask in the NTCIR Math Task. Participants have received the NTCIR-MIR dataset which contains 100 000 XHTML full texts of articles from the arXiv. Formulae are marked up as MathML (presentation markup with annotated content markup and LATEX source). 1.1 Subtasks The MIR Subtask in NTCIR-10 has three challenges; queries are given in chapters 2-4 below, the format is Formula Search (Automated) Participating IR systems obtain a list of queries consisting of formulae (possibly) with wildcards (query variables) and return for every query an ordered list of XPointer identifiers of formulae claimed to match the query, plus possible supporting evidence (e.g. a substitution for query variables). Full-Text Search (Automated) This is like formula search above, only that IR results are ordered lists of \hits" (i.e. XPointer references into the documents with a highlighted result fragments plus supporting evidence1). EdN:1 Open Information Retrieval (Semi-Automated) In contrast to the first two challenges, where the systems are run in batch-mode (i.e. without human intervention), in this one mathe- maticians will challenge the (human) participants to find specific information in a document corpus via human-readable descriptions (natural language text), which are translated by the participants to their IR systems. -
THE DIMENSION of the TORELLI GROUP 1. Introduction Let S Be A
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 23, Number 1, January 2010, Pages 61–105 S 0894-0347(09)00643-2 Article electronically published on July 10, 2009 THE DIMENSION OF THE TORELLI GROUP MLADEN BESTVINA, KAI-UWE BUX, AND DAN MARGALIT 1. Introduction Let S be a surface (unless specified otherwise, we take all surfaces to be con- nected, orientable, and of finite type). Let Mod(S)bethemapping class group of S, + + defined as π0(Homeo (S)), where Homeo (S) is the group of orientation-preserving homeomorphisms of S.TheTorelli group I(S) is the kernel of the natural action of Mod(S)onH1(S, Z). When S is Sg, the closed surface of genus g, this action is symplectic (it preserves the algebraic intersection number), and it is a classical fact that Mod(Sg) surjects onto Sp(2g, Z). All of this information is encoded in the following short exact sequence: 1 −→I(Sg) −→ Mod(Sg) −→ Sp(2g, Z) −→ 1. Cohomological dimension. For a group G,wedenotebycd(G)itscohomological dimension, which is the supremum over all n so that there exists a G-module M with Hn(G, M) =0. Theorem A. For g ≥ 2, we have cd(I(Sg)) = 3g − 5. ∼ Since Mod(S0)=1,wehaveI(S0) = 1. Also, it is a classical fact that Mod(S1) = Sp(2, Z)=SL(2, Z), and so I(S1)isalsotrivial. The lower bound for Theorem A was already given by Mess [36] in an unpublished paper from 1990. Theorem 1.1 (Mess). For g ≥ 2, we have cd(I(Sg)) ≥ 3g − 5. -
Partial Euler Characteristic, Normal Generations and the Stable D (2
Partial Euler Characteristic, Normal Generations and the stable D(2) problem Feng Ji and Shengkui Ye October 10, 2018 Abstract We study the interplay among Wall’s D(2) problem, normal generation conjecture (the Wiegold Conjecture) of perfect groups and Swan’s problem on partial Euler characteristic and deficiency of groups. In particular, for a 3-dimensional complex X of cohomological dimension 2 with finite fundamental group, assuming the Wiegold conjecture holds, we prove that X is homotopy equivalent to a finite 2-complex after wedging a copy of 2 sphere S . 1 Introduction In this article, we study several classical problems in low-dimensional homotopy theory and group theory, focusing on the interplay among these problems. Let us first recall the following Swan’s problem. Let G be a group and Z G the group ring. Swan [16] defines the partial Euler characteristic µn(G) as follows. Let F be a resolution ···→ F2 → F1 → F0 → Z → 0 of the trivial ZG-module Z, in which each Fi is ZG-free on fi generators. For an integer n ≥ 0, if f0,f1,f2, ··· ,fn arXiv:1503.01987v2 [math.AT] 16 Jan 2018 are finite, define n µn(F )= fn − fn−1 + fn−2 − · · · + (−1) f0. If there exists a resolution F such that µn(F ) is defined, we let µn(G) be the infimum of µn(F ) over all such resolutions F. We call the truncated free resolu- tion Fn →···→ F1 → F0 → Z → 0 an algebraic n-complex if each Fi is finitely generated as a ZG-module (following the terminology of Johnson [8]). -
The Tits Alternative
The Tits Alternative Matthew Tointon April 2009 0 Introduction In 1972 Jacques Tits published his paper Free Subgroups in Linear Groups [Tits] in the Journal of Algebra. Its key achievement was to prove a conjecture of H. Bass and J.-P. Serre, now known as the Tits Alternative for linear groups, namely that a finitely-generated linear group over an arbitrary field possesses either a solvable subgroup of finite index or a non-abelian free subgroup. The aim of this essay is to present this result in such a way that it will be clear to a general mathematical audience. The greatest challenge in reading Tits's original paper is perhaps that the range of mathematics required to understand the theorem's proof is far greater than that required to understand its statement. Whilst this essay is not intended as a platform in which to regurgitate theory it is very much intended to overcome this challenge by presenting sufficient background detail to allow the reader, without too much effort, to enjoy a proof that is pleasing in both its variety and its ingenuity. Large parts of the prime-characteristic proof follow basically the same lines as the characteristic-zero proof; however, certain elements of the proof, particularly where it is necessary to introduce field theory or number theory, can be made more concrete or intuitive by restricting to characteristic zero. Therefore, for the sake of clarity this exposition will present the proof over the complex numbers, although where clarity and brevity are not impaired by considering a step in the general case we will do so. -
Amenable Groups and Groups with the Fixed Point Property
AMENABLE GROUPS AND GROUPS WITH THE FIXED POINT PROPERTY BY NEIL W. RICKERTÍ1) Introduction. A left amenable group (i.e., topological group) is a group with a left invariant mean. Roughly speaking this means that the group has a finitely additive set function of total mass 1, which is invariant under left translation by elements of the group. A precise definition will be given later. Groups of this type have been studied in [2], [5], [7], [8], [12], [13], and [20]. Most of the study to date has been concerned with discrete groups. In this paper we shall attempt to extend to locally compact groups much of what is known for the discrete case. Another interesting class of groups consists of those groups which have the fixed point property. This is the name given by Furstenberg to groups which have a fixed point every time they act affinely on a compact convex set in a locally convex topological linear space. Day [3] has shown that amenability implies the fixed point property. For discrete groups he has shown the converse. Along with amenable groups, we shall study, in this paper, groups with the fixed point property. This paper is based on part of the author's Ph.D. dissertation at Yale University. The author wishes to express his thanks to his adviser, F. J. Hahn. Notation. Group will always mean topological group. For a group G, G0 will denote the identity component. Likewise H0 will be the identity component of H, etc. Banach spaces, topological vector spaces, etc., will always be over the real field. -
The Boundary and the Virtual Cohomological Dimension of Coxeter Groups
Houston Journal of Mathematics c 2000 University of Houston Volume 26, No. 4, 2000 THE BOUNDARY AND THE VIRTUAL COHOMOLOGICAL DIMENSION OF COXETER GROUPS TETSUYA HOSAKA AND KATSUYA YOKOI Communicated by Charles Hagopian Abstract. This paper consists of three parts: 1) We give some properties about the virtual cohomological dimension of Coxeter groups over principal ideal domains. 2) For a right-angled Coxeter group Γ with vcdR Γ = n, we construct a sequence ΓW0 ΓW1 ΓWn 1 of parabolic subgroups º º ¡ ¡ ¡ º − with vcdR ΓWi = i. 3) We show that a parabolic subgroup of a right-angled Coxeter group is of ¬nite index if and only if their boundaries coincide. 1. Introduction and Preliminaries The purpose of this paper is to study Coxeter groups and their boundaries. Let V be a ¬nite set and m : V V N a function satisfying the following ¢ ! [f1g conditions: (1) m(v; w) = m(w; v) for all v; w V , 2 (2) m(v; v) = 1 for all v V , and 2 (3) m(v; w) 2 for all v = w V . ≥ 6 2 A Coxeter group is a group Γ having the presentation V (vw)m(v;w) = 1 for v; w V ; h j 2 i where if m(v; w) = , then the corresponding relation is omitted, and the pair 1 (Γ;V ) is called a Coxeter system. If m(v; w) = 2 or for all v = w V , then 1 6 2 (Γ;V ) is said to be right-angled. For a Coxeter system (Γ;V ) and a subset W V , º 1991 Mathematics Subject Classi¬cation. -
No Tits Alternative for Cellular Automata
No Tits alternative for cellular automata Ville Salo∗ University of Turku January 30, 2019 Abstract We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alter- native. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group Z) and the one-sided case (spatially acting monoid N, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sofic shifts, automorphism groups of multidimensional sub- shifts of finite type with positive entropy and dense minimal points, auto- morphism groups of full shifts over non-periodic groups, and the mapping class groups of two-sided one-dimensional transitive SFTs. We also show that the classical Tits alternative applies to one-dimensional (multi-track) reversible linear cellular automata over a finite field. 1 Introduction In [52] Jacques Tits proved that if F is a field (with no restrictions on charac- teristic), then a finitely-generated subgroup of GL(n, F ) either contains a free group on two generators or contains a solvable subgroup of finite index. We say that a group G satisfies the Tits alternative if whenever H is a finitely-generated subgroup of G, either H is virtually solvable or H contains a free group with two generators. Whether an infinite group satisfies the Tits alternative is one of the natural questions to ask.