Relevant bibliographies by topics / Ergodic and geometric group theory

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Ergodic and geometric group theory'

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Published: 31 August 2024

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Journal articles on the topic "Ergodic and geometric group theory"

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Skripchenko, Alexandra Sergeevna. "Renormalization in one-dimensional dynamics." Russian Mathematical Surveys 78, no. 6 (2023): 983–1021. http://dx.doi.org/10.4213/rm10110e.

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The study of the dynamical and topological properties of interval exchange transformations and their natural generalizations is an important problem, which lies at the intersection of several branches of mathematics, including dynamical systems, low-dimensional topology, algebraic geometry, number theory, and geometric group theory. The purpose of the survey is to make a systematic presentation of the existing results on the ergodic and geometric characteristics of the one-dimensional maps under consideration, as well as on the measured foliations on surfaces and two-dimensional complexes that one can associate with these maps. These results are based on the research of the ergodic properties of the renormalization process - an algorithm that takes an original dynamical system and builds a sequence of equivalent dynamical systems with a smaller support set. For all dynamical systems considered in the paper these renormalization algorithms can be viewed as multidimensional fraction algorithms. Bibiliography: 74 titles.
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Hartman, Yair, and Ariel Yadin. "Furstenberg entropy of intersectional invariant random subgroups." Compositio Mathematica 154, no. 10 (September 17, 2018): 2239–65. http://dx.doi.org/10.1112/s0010437x18007261.

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We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply, for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.
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Guirardel, Vincent, Camille Horbez, and Jean Lécureux. "Cocycle superrigidity from higher rank lattices to $ {{\rm{Out}}}{(F_N)} $." Journal of Modern Dynamics 18 (2022): 291. http://dx.doi.org/10.3934/jmd.2022010.

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<p style='text-indent:20px;'>We prove a rigidity result for cocycles from higher rank lattices to <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{Out}(F_N) $\end{document}</tex-math></inline-formula> and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula> be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let <inline-formula><tex-math id="M4">\begin{document}$ G \curvearrowright X $\end{document}</tex-math></inline-formula> be an ergodic measure-preserving action on a standard probability space, and let <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> be a torsion-free hyperbolic group. We prove that every Borel cocycle <inline-formula><tex-math id="M6">\begin{document}$ G\times X\to \mathrm{Out}(H) $\end{document}</tex-math></inline-formula> is cohomologous to a cocycle with values in a finite subgroup of <inline-formula><tex-math id="M7">\begin{document}$ \mathrm{Out}(H) $\end{document}</tex-math></inline-formula>. This provides a dynamical version of theorems of Farb–Kaimanovich–Masur and Bridson–Wade asserting that every homomorphism from <inline-formula><tex-math id="M8">\begin{document}$ G $\end{document}</tex-math></inline-formula> to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image.</p><p style='text-indent:20px;'>The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.</p>
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Kida, Yoshikata. "Ergodic group theory." Sugaku Expositions 35, no. 1 (April 7, 2022): 103–26. http://dx.doi.org/10.1090/suga/470.

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Young, Lai-Sang. "Geometric and Ergodic Theory of Hyperbolic Dynamical Systems." Current Developments in Mathematics 1998, no. 1 (1998): 237–78. http://dx.doi.org/10.4310/cdm.1998.v1998.n1.a6.

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Orponen, Tuomas, Pablo Shmerkin, and Hong Wang. "Incidence Problems in Harmonic Analysis, Geometric Measure Theory, and Ergodic Theory." Oberwolfach Reports 20, no. 2 (December 21, 2023): 1397–452. http://dx.doi.org/10.4171/owr/2023/25.

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Clay, Matt. "Geometric Group Theory." Notices of the American Mathematical Society 69, no. 10 (November 1, 2022): 1. http://dx.doi.org/10.1090/noti2572.

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Ziegler, Tamar. "An application of ergodic theory to a problem in geometric ramsey theory." Israel Journal of Mathematics 114, no. 1 (December 1999): 271–88. http://dx.doi.org/10.1007/bf02785583.

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BESSA, MÁRIO, and JORGE ROCHA. "Contributions to the geometric and ergodic theory of conservative flows." Ergodic Theory and Dynamical Systems 33, no. 6 (August 22, 2012): 1709–31. http://dx.doi.org/10.1017/etds.2012.110.

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AbstractWe prove the following dichotomy for vector fields in a $C^1$-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and $C^1$-stably ergodic flow can be $C^1$-approximated by another volume-preserving flow which is non-uniformly hyperbolic.
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BOWEN, LEWIS, and AMOS NEVO. "Hyperbolic geometry and pointwise ergodic theorems." Ergodic Theory and Dynamical Systems 39, no. 10 (December 12, 2017): 2689–716. http://dx.doi.org/10.1017/etds.2017.128.

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We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real rank one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.
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Dissertations / Theses on the topic "Ergodic and geometric group theory"

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Cannizzo, Jan. "Schreier Graphs and Ergodic Properties of Boundary Actions." Thesis, Université d'Ottawa / University of Ottawa, 2014. http://hdl.handle.net/10393/31444.

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This thesis is broadly concerned with two problems: investigating the ergodic properties of boundary actions, and investigating various properties of Schreier graphs. Our main result concerning the former problem is that, in a variety of situations, the action of an invariant random subgroup of a group G on a boundary of G (e.g. the hyperbolic boundary, or the Poisson boundary) is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda and establishes a connection between invariant random subgroups and normal subgroups. We approach the latter problem from a number of directions (in particular, both in the presence and the absence of a probability measure), with an emphasis on what we term Schreier structures (edge-labelings of a given graph which turn it into a Schreier coset graph). One of our main results is that, under mild assumptions, there exists a rich space of invariant Schreier structures over a given unimodular graph structure, in that this space contains uncountably many ergodic measures, many of which we are able to describe explicitly.
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Long, Yusen. "Diverse aspects of hyperbolic geometry and group dynamics." Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASM016.

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Cette thèse explore divers sujets liés à la géométrie hyperbolique et à la dynamique de groupes, dans le but d'étudier l'interaction entre la géométrie et la théorie de groupes. Elle couvre un large éventail de disciplines mathématiques, telles que la géométrie convexe, l'analyse stochastique, la théorie ergodiques et géométriques de groupes, et la topologie en basses dimensions, et cætera. Comme résultats de recherche, la géométrie hyperbolique des corps convexes en dimension infinie est examinée en profondeur, et des tentatives sont faites pour développer la géométrie intégrale en dimension infinie d'un point de vue de l'analyse stochastique. L'étude des gros groupes de difféotopies, un sujet d'actualité en topologie en basses dimensions et en théorie géométrique de groupes, est entreprise avec une détermination complète de leur propriété de point fixe sur les compacts. La thèse étudie la connexité du bord de Gromov des graphes de courbes fins, un outil combinatoire utilisé dans l'étude des groupes d'homéomorphismes des surfaces de type fini. Enfin, la thèse clarifie également certains théorèmes folkloriques concernant les espaces hyperboliques au sens de Gromov et la dynamique des groupes moyennables sur ces espaces
This thesis explores diverse topics related to hyperbolic geometry and group dynamics, aiming to investigate the interplay between geometry and group theory. It covers a wide range of mathematical disciplines, such as convex geometry, stochastic analysis, ergodic and geometric group theory, and low-dimensional topology, etc. As research outcomes, the hyperbolic geometry of infinite-dimensional convex bodies is thoroughly examined, and attempts are made to develop integral geometry in infinite dimensions from a perspective of stochastic analysis. The study of big mapping class groups, a current focus in low-dimensional topology and geometric group theory, is undertaken with a complete determination of their fixed-point on compacta property. The thesis also clarifies certain folklore theorems regarding the Gromov hyperbolic spaces and the dynamics of amenable groups on them. Last but not the least, the thesis studies the connectivity of the Gromov boundary of fine curve graphs, a combinatorial tool employed in the study of the homeomorphism groups of surfaces of finite type
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Benson, Martin. "Topics in geometric group theory." Thesis, University of Nottingham, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.428957.

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Williams, Benjamin Thomas. "Two topics in geometric group theory." Thesis, University of Southampton, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.323942.

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Isenrich, Claudio Llosa. "Kähler groups and Geometric Group Theory." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:4a7ab097-4de5-4b72-8fd6-41ff8861ffae.

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In this thesis we study Kähler groups and their connections to Geometric Group Theory. This work presents substantial progress on three central questions in the field: (1) Which subgroups of direct products of surface groups are Kähler? (2) Which Kähler groups admit a classifying space with finite (n-1)-skeleton but no classifying space with finitely many n-cells? (3) Is it possible to give explicit finite presentations for any of the groups constructed in response to Question 2? Question 1 was raised by Delzant and Gromov. Question 2 is intimately related to Question 1: the non-trivial examples of Kähler subgroups of direct products of surface groups never admit a classifying space with finite skeleton. The only known source of non-trivial examples for Questions 1 and 2 are fundamental groups of fibres of holomorphic maps from a direct product of closed surfaces onto an elliptic curve; the first such construction is due to Dimca, Papadima and Suciu. Question 3 was posed by Suciu in the context of these examples. In this thesis we: provide the first constraints on Kähler subdirect products of surface groups (Theorem 7.3.1); develop new construction methods for Kähler groups from maps onto higher-dimensional complex tori (Section 6.1); apply these methods to obtain irreducible examples of Kähler subgroups of direct products of surface groups which arise from maps onto higher-dimensional tori and use them to show that our conditions in Theorem 7.3.1 are minimal (Theorem A); apply our construction methods to produce irreducible examples of Kähler groups that (i) have a classifying space with finite (n-1)-skeleton but no classifying space with finite n-skeleton and (ii) do not have a subgroup of finite index which embeds in a direct product of surface groups (Theorem 8.3.1); provide a new proof of Biswas, Mj and Pancholi's generalisation of Dimca, Papadima and Suciu's construction to more general maps onto elliptic curves (Theorem 4.3.2) and introduce invariants that distinguish many of the groups obtained from this construction (Theorem 4.6.2); and, construct explicit finite presentations for Dimca, Papadima and Suciu's groups thereby answering Question 3 (Theorem 5.4.4)).
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Ashdown, M. A. J. "Geometric algebra, group theory and theoretical physics." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.596181.

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This dissertation applies the language of geometric algebra to group theory and theoretical physics. Geometric algebra, which is introduced in Chapter 2, provides a natural extension of the concept of multiplication from real numbers to geometric objects such as line segments and planes. It is based on Clifford algebra and augmented by auxiliary definitions which give it a geometric interpretation. Since geometric algebra provides a natural encoding of the concepts of directed quantities, it has the potential to unify many of the disparate systems of notation that are used in mathematics. In Chapter 3, the properties of multilinear functions are investigated and the theory is developed to make them useful for formulating the representation of groups. It will be found that multilinear functions are more flexible than their tensor or matrix counterparts in traditional linear algebra. Multilinear functions can be classified according to the symmetry class of their arguments and their behaviour under the monogenic or harmonic decomposition. It is found that the previous definitions of monogenic and harmonic functions need some modification if they are to be defined consistently. Polynomial projection is also discussed, a technique that is useful in constructing non-linear functions from linear functions, an operation outside the scope of conventional linear algebra. In Chapter 4, multilinear functions are used to construct the irreducible representations of the three regular classes of classical groups; rotation groups, the special unitary and special linear group, and the symplectic group. In each case it is found that a decomposition must be applied to the multilinear functions in order to find the irreducible representations of the groups. For the representations of some of the groups this entails finding the harmonic or monogenic parts of the functions. The groups can be realised as subgroups of the spin group of some dimension and signature. However, geometric algebra provides such a rich algebraic structure that the representations of the groups can be realised in more than one way. In Chapter 7 a brief review is given of computer software for performing symbolic calculations with geometric algebra. A new software package which performs semi-symbolic manipulation of multivectors in spaces of any dimension and signature is presented.
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Gill, Olivia Jo. "Geometric and homological methods in group theory : constructing small group resolutions." Thesis, London Metropolitan University, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.573402.

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Given two groups K and H for which we have the free crossed resolutions, B* ɛ K and C* ɛ H respectively. Our aim is to construct a free crossed resolution, A* ɛ G, by way of induction on the degree n, for any semidirect product G = K >
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Joubert, Paul. "Geometric actions of the absolute Galois group." Thesis, Stellenbosch : University of Stellenbosch, 2006. http://hdl.handle.net/10019.1/2508.

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Thesis (MSc (Mathematics))--University of Stellenbosch, 2006.
This thesis gives an introduction to some of the ideas originating from A. Grothendieck's 1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new geometric approach to studying the absolute Galois group over the rationals by considering its action on certain geometric objects such as dessins d'enfants (called stick figures in this thesis) and the fundamental groups of certain moduli spaces of curves. I start by defining stick figures and explaining the connection between these innocent combinatorial objects and the absolute Galois group. I then proceed to give some background on moduli spaces. This involves describing how Teichmuller spaces and mapping class groups can be used to address the problem of counting the possible complex structures on a compact surface. In the last chapter I show how this relates to the absolute Galois group by giving an explicit description of the action of the absolute Galois group on the fundamental group of a particularly simple moduli space. I end by showing how this description was used by Y. Ihara to prove that the absolute Galois group is contained in the Grothendieck-Teichmuller group.
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El-Mosalamy, Mohamed Soliman Hassan. "Applications of star complexes in group theory." Thesis, University of Glasgow, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.293464.

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Fennessey, Eric James. "Some applications of geometric techniques in combinatorial group theory." Thesis, University of Glasgow, 1989. http://theses.gla.ac.uk/6159/.

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Combinatorial group theory abounds with geometrical techniques. In this thesis we apply some of them to three distinct areas. In Chapter 1 we present all of the techniques and background material neccessary to read chapters 2,3,4. We begin by defining complexes with involutary edges and define coverings of these. We then discuss equivalences between complexes and use these in §§1.3 and 1.4 to give a way (the level method) of simplifying complexes and an application of this method (Theorem 1.3). We then discuss star-complexes of complexes. Next we present background material on diagrams and pictures. The final section in the chapter deals with SQ-universality. The.basic discussion of complexes is taken from notes, by Pride, on complexes without involutary edges, and modified by myself to cover complexes with involution. Chapters 2,3, and 4 are presented in the order that the work for them was done. Chapters 2,3, alld 4 are intended (given the material in chapter 1) to be self contained, and (iv) each has a full introduction. In Chapter 2 we use diagrams and pictures to study groups with the following structure. (a) Let r be a graph with vertex set V and edge set E. We assume that no vertex of r is isolated. (b) For each vertex VEV there is a non-trivial group Gv ' (c) For each edge e-{u,v}EE there is a set Se of cyclically reduced elements of Gu*Gv , each of length at least two. We define Ge to be the quotient of Gu*Gv by the normal closure of Se. We let G be the quotient of *Gv by the normal closure of VEV S- USe. For convenience, we write eEE The above is a generalization ofa situation studied by Pride [35], where each Gv was infinite cyclic.' Let e-{u,v} be an edge of r. We will say that Ge has property-Wk if no non-trivial element of Gu*Gv of free product length less than or equal to 2k is in the kernel of the natural epimorphism (v) We will work with one of the following: (I) Each Ge has property-W2 (II) r is triangle-free and each Ge has property-WI' Assuming that (I) or (II) holds we: (i) prove a Freihietssatz for these groups; (ii) give sufficient conditions for the groups to be SQ-universal; (iii) prove a result which allows us to give long exact sequences relating the (co)-homology G to the (co)-homology of the groups The work in Chapter 2 is in some senses the least original. The proofs are extensions of proofs given in [35] and [39] for the case when each Gv is infinite cyclic. However. there are some technical difficulties which we had to overcome. In chapter 3 we use the two ideas of star-complexes and coverings to look at NEC-groups. An NEC (Non-Euclidean Crystallographic) group is a discontinuous group of isometries (some of which may be (vi) orientation reversing) of the Non-Euclidean plane. According to Yilkie [46], a finitely generated NEC-group with compact orbit space has a presentation as follows: Involutary generators: Yij (i,j)EZo Non-involutary generators: 6i (iElf), tk (l~~r) (*) Defining paths: (YijYij+,)mij (iElf, l~j~n(i)-l) where In Hoare, Karrass and Solitar [22] it is shown that a subgroup of finite index in a group with a presentation of the form (*), has itself a presentation of the form (*). In [22] the same authors show that a subgroup of infinite ingex in a group with a presentation of the form (*) is a free product of groups of the following types: (A) Cyclic groups. (vii) (B) Groups with presentations of the form Xl' ... 'Xn involutary. (e) Groups with presentations of the form Xi (iEZ) involutary. We define what we mean by an NEe-complex. (This involves a structural re$triction on the form of the star-complex of the complex.) It is obvious from the definition that this class of complexes is clo$ed under coverings, so that the class of fundamental groups of NEe-complexes is trivially closed under taking subgroups. We then obtain structure theorems for both finite and infinite NEe-complexes. We show that the fundamental group of a finite NEe-complex has a presentation of the form (*) and that the fundamental group of an infinite NEe-complex is a free product of groups of the forms (A). (B) and (e) above. We then use coverings to derive some of the results on normal subgroups of NEe-groups given in [5] and [6]. , (viii) In chapter 4 we use the techniques of coverings and diagrams. to stue,iy the SQ-universau'ty of Coxeter groups. This is a problem due to B.H. Neumann (unpublished). see [40]. A Coxeter pair is a 2-tup1e (r.~) where r is a graph (with vertex set V(r) and edge set E(r» and ~ is a map from E(r) to {2.3.4 •.•• }. We associate with (r.~) the Coxeter group c(r,~) defined by the presentation tr(r,~)-, where each generator is involutary. Following Appel and Schupp [1] we say that a Coxeter pair is of large type if 2/Im~. I conjecture that if (r,~) is of large type with IV(r)I~3 and r not a triangle with all edges mapped to 3 by ~. then C(r,~) is SQ-universa1. In connection with this conjecture we firstly prove (Theorem 4.1), Let (r,~) be a Coxeter pair of large type. Suppose (A) r is incomplete on at least three vertices, or (B) r is complete on at least five vertices and for 1 < - 2 (ix) Then C(r,~) is SQ-universal. Secondly we prove a result (Theorem 4.2) which shows: If (r,~) is a Coxeter pair with IV(r)I~4 and hcf[~(E(r»] > 1, then C(r,~) is either SQ-universal or is soluble of length at most three. Moreover our Theorem allows us to tell the two possibilities apart. The proof of this result leads to consideration of the following question: If a direct sum of groups is SQ-universal, does this imply that one of the summands is itself SQ-universal? We show (in appendix B) that the answer is "yes" for countable direct sums. We consider the results in chapter 4 and its appendix to be the most significant part of this thesis
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Books on the topic "Ergodic and geometric group theory"

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Koli︠a︡da, S. F. Dynamics and numbers: A special program, June 1-July 31, 2014, Max Planck Institute for Mathematics, Bonn, Germany : international conference, July 21-25, 2014, Max Planck Institute for Mathematics, Bonn, Germany. Edited by Max-Planck-Institut für Mathematik. Providence, Rhode Island: American Mathematical Society, 2016.

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Burger, Marc. Rigidity in Dynamics and Geometry: Contributions from the Programme Ergodic Theory, Geometric Rigidity and Number Theory, Isaac Newton Institute for the Mathematical Sciences Cambridge, United Kingdom, 5 January - 7 July 2000. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002.

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Bhattacharya, Siddhartha, Tarun Das, Anish Ghosh, and Riddhi Shah. Recent trends in ergodic theory and dynamical systems: International conference in honor of S.G. Dani's 65th birthday, December 26--29, 2012, Vadodara, India. Providence, Rhode Island: American Mathematical Society, 2015.

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Bestvina, Mladen, Michah Sageev, and Karen Vogtmann. Geometric group theory. Providence, RI: American Mathematical Society, 2014.

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Charney, Ruth, Michael Davis, and Michael Shapiro, eds. Geometric Group Theory. Berlin, New York: DE GRUYTER, 1995. http://dx.doi.org/10.1515/9783110810820.

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Arzhantseva, Goulnara N., José Burillo, Laurent Bartholdi, and Enric Ventura, eds. Geometric Group Theory. Basel: Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8412-8.

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Löh, Clara. Geometric Group Theory. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-72254-2.

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Dani, S. G., and Anish Ghosh, eds. Geometric and Ergodic Aspects of Group Actions. Singapore: Springer Singapore, 2019. http://dx.doi.org/10.1007/978-981-15-0683-3.

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Zimmer, Robert J. Ergodic theory, groups, and geometry. Providence, R.I: American Mathematical Society, 2008.

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Doran, Robert S., Calvin C. Moore, and Robert J. Zimmer, eds. Group Representations, Ergodic Theory, and Mathematical Physics. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/conm/449.

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Book chapters on the topic "Ergodic and geometric group theory"

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Polterovich, Leonid. "An Application to Ergodic Theory." In The Geometry of the Group of Symplectic Diffeomorphism, 83–87. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8299-6_11.

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Lyndon, Roger C., and Paul E. Schupp. "Geometric Methods." In Combinatorial Group Theory, 114–73. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/978-3-642-61896-3_3.

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Löh, Clara. "Group actions." In Geometric Group Theory, 75–114. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-72254-2_4.

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Guirardel, Vincent. "Geometric small cancellation." In Geometric Group Theory, 55–90. Providence, Rhode Island: American Mathematical Society, 2014. http://dx.doi.org/10.1090/pcms/021/03.

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Baumgartner, Udo. "Totally Disconnected, Locally Compact Groups as Geometric Objects." In Geometric Group Theory, 1–20. Basel: Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8412-8_1.

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Miasnikov, Alexei, Enric Ventura, and Pascal Weil. "Algebraic Extensions in Free Groups." In Geometric Group Theory, 225–53. Basel: Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8412-8_12.

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Ceccherini-Silberstein, Tullio, and Michel Coornaert. "On the Surjunctivity of Artinian Linear Cellular Automata over Residually Finite Groups." In Geometric Group Theory, 37–44. Basel: Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8412-8_3.

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de Cornulier, Yves, and Avinoam Mann. "Some Residually Finite Groups Satisfying Laws." In Geometric Group Theory, 45–50. Basel: Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8412-8_4.

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de Cornulier, Yves, and Pierre de la Harpe. "Décompositions de Groupes par Produit Direct et Groupes de Coxeter." In Geometric Group Theory, 75–102. Basel: Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8412-8_7.

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Houcine, Abderezak Ould. "Limit Groups of Equationally Noetherian Groups." In Geometric Group Theory, 103–19. Basel: Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8412-8_8.

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Conference papers on the topic "Ergodic and geometric group theory"

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Ruelle, David. "Ergodic Theory of Chaos." In Optical Bistability. Washington, D.C.: Optica Publishing Group, 1985. http://dx.doi.org/10.1364/obi.1985.wc1.

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Determinsistic chaos arises in a variety of nonlinear dynamical systems in physics, and in particular in optics. One has now gained a reasonable understanding of the onset of chaos in terms of the geometry of bifurcations and strange attractors. This geometric approach does not work for attractors of more than two or three dimensions. For these, however, ergodic theory provides new concepts: characteristic exponents, entropy, information dimension, which are reproducibly estimated from physical experiments. The Characteristic exponents measure the rate of divergence of nearby trajectories of a dynamical system, the entropy measures the rate of information creation by the system, and the information dimension is a fractal dimension of particular interest. There are inequalities (or even identities) relating the entropy and information dimension to the characteristic exponents. The experimental measure of these ergodic quantities provides a numerical estimate of the instability of chaotic systems, and of the number of "degrees of freedom" which they possess.
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Berendsohn, Benjamin Aram, and Laszlo Kozma. "Group Testing with Geometric Ranges." In 2022 IEEE International Symposium on Information Theory (ISIT). IEEE, 2022. http://dx.doi.org/10.1109/isit50566.2022.9834574.

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BOEIRA DORNELAS, BIANCA, and FRANCESCO MATUCCI. "Introduction to Combinatorial and Geometric Group Theory." In XXV Congresso de Iniciação Cientifica da Unicamp. Campinas - SP, Brazil: Galoa, 2017. http://dx.doi.org/10.19146/pibic-2017-79172.

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Ruiz S., Oscar E., and Placid M. Ferreira. "Algebraic geometry and group theory in geometric constraint satisfaction." In the international symposium. New York, New York, USA: ACM Press, 1994. http://dx.doi.org/10.1145/190347.190421.

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Wolf, Kurt Bernardo. "Introduction to Lie geometric optics." In The XXX Latin American school of physics ELAF: Group theory and its applications. AIP, 1996. http://dx.doi.org/10.1063/1.50229.

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Peng, Bo. "An approach to group decision making based on interval-valued intuitionistic fuzzy geometric distance measures." In 2015 International Conference on Fuzzy Theory and Its Applications (iFUZZY). IEEE, 2015. http://dx.doi.org/10.1109/ifuzzy.2015.7391901.

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Clayton, John D. "Shock compression of metal single crystals modeled via Finsler-geometric continuum theory." In SHOCK COMPRESSION OF CONDENSED MATTER - 2017: Proceedings of the Conference of the American Physical Society Topical Group on Shock Compression of Condensed Matter. Author(s), 2018. http://dx.doi.org/10.1063/1.5045034.

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Gou, J. B., Y. X. Chu, H. Wu, and Z. X. Li. "A Geometric Theory for Formulation of Form, Profile and Orientation Tolerances: Problem Formulation." In ASME 1998 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/detc98/dfm-5743.

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Abstract This paper develops a geometric theory which unifies the formulation and evaluation of form (straightness, flatness, cylindricity and circularity), profile and orientation tolerances stipulated in ANSI Y14.5M standard. In the paper, based on an an important observation that a toleranced feature exhibits a symmetry subgroup G0 under the action of the Euclidean group, SE(3), we identify the configuration space of a toleranced (or a symmetric) feature with the homogeneous space SE(3)/G0 of the Euclidean group. Geometric properties of SE(3)/G0, especially its exponential coordinates carried over from that of SE(3), are analyzed. We show that all cases of form, profile and orientation tolerances can be formulated as a minimization or constrained minimization problem on the space SE(3)/G0, with G0 being the symmetry subgroup of the underlying feature. We transform the non-differentiable minimization problem into a differentiable minimization problem over an extended configuration space. Using geometric properties of SE(3)/G0, we derive a sequence of linear programming problems whose solutions can be used to approximate the minimum zone solutions.
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Jung, Tae-Hwa, and Changhoon Lee. "Supercritical Group Velocity for Dissipative Waves in Shallow Water." In ASME 2012 31st International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/omae2012-83279.

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The group velocity for waves with energy dissipation in shallow water was investigated. In the Eulerian viewpoint, the geometric optics approach was used to get, at the first order, complex-valued wave numbers from given real-valued angular frequency, water depth, and damping coefficient. The phase velocity was obtained as the ratio of angular frequency to realvalued wave number. Then, at the second order, we obtained the energy transport equation which gives the group velocity. We also used the Lagrangian geometric optics approach which gives complex-valued angular frequencies from real-valued wave number, water depth, and damping coefficient. A noticeable thing was found that the group velocity is always greater than the phase velocity (i.e., supercritical group velocity) in the presence of energy dissipation which is opposite to the conventional theory for non-dissipative waves. The theory was proved through numerical experiments for dissipative bichromatic waves which propagate on a horizontal bed. Both the wave length and wave energy decrease for waves with energy dissipation. As a result, wave transformation such as shoaling, refraction, and diffraction are all affected by the energy dissipation. This implies that the shoaling, refraction, and diffraction coefficients for dissipative waves are different from the corresponding coefficients for non-dissipative waves. The theory was proved through numerical experiments for dissipative monochromatic waves which propagate normally or obliquely on a planar slope.
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Li, Z. X., B. Kang, J. B. Gou, Y. X. Chu, and M. Yeung. "Fundamentals of Workpiece Localization: Theory and Algorithms." In ASME 1996 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/imece1996-0811.

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Abstract In this paper, we present an algebraic algorithm for workpiece localization. First, we formulate the problem as a least-square problem in the configuration space Q = SE(3) × ℝ3n, where SE(3) is the Euclidean group, and n is the number of measurement points to be matched by corresponding home surface points of the workpiece. Then, we use the geometric properties of the Euclidean group to compute for the critical points of the objective function. Doing so we derive an algebraic formula for the optimal Euclidean transformation in terms of the measurement points and the corresponding home surface points. We also give for each measurement point a system of two nonlinear equations from which the corresponding home surface point nearest to the measurement point can be solved. Finally, based on these analytic results we present an iterative algorithm for obtaining the complete solution of the least-square problem.
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