DOCSLIB.ORG

Tree-Graded Spaces and Asymptotic Cones of Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tree-Graded Spaces and Asymptotic Cones of Groups Tree-graded spaces and asymptotic cones of groups Cornelia Drut»u and Mark Sapir¤ with an Appendix by Denis Osin and Mark Sapir Abstract We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to ¯nd geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the ¯rst example of ¯nitely generated group with a continuum of non-¼1-equivalent asymptotic cones. Note that by a result of Kramer, Shelah, Tent and Thomas, continuum is the maximal possible number of di®erent asymptotic cones of a ¯nitely generated group, provided that the Continuum Hypothesis is true. Contents 1 Introduction 2 1.1 Open questions . 10 1.2 Plan of the paper . 10 2 Tree-graded spaces 11 2.1 Properties of tree-graded spaces . 11 2.2 Modifying the set of pieces . 16 2.3 Geodesics in tree-graded spaces . 18 2.4 Cut-points and tree-graded spaces . 19 3 Ultralimits and asymptotic cones 20 3.1 Preliminaries . 20 3.2 Ultralimits of asymptotic cones are asymptotic cones . 23 3.3 Another de¯nition of asymptotic cones . 25 3.4 Simple triangles in ultralimits of metric spaces . 25 4 A characterization of asymptotically tree-graded spaces 31 5 Quasi-isometric behavior 43 5.1 Asymptotically tree-graded spaces . 43 5.2 Asymptotically tree-graded groups . 45 ¤The ¯rst author is grateful to the CNRS of France for granting her the d¶el¶egationCNRS status in the 2003- 2004 academic year. The research of the second author was supported in part by the NSF grant DMS 0072307 and by the US-Israeli BSF grant 1999298. We worked on the present paper during the ¯rst author's visits to Vanderbilt University, and the second author's visits to University of Lille-1 and to the IHES. The authors are grateful to the administration and faculty of these institutions for their support and hospitality. 1 6 Cut-points in asymptotic cones of groups 49 6.1 Groups with central in¯nite cyclic subgroups . 49 6.2 Groups satisfying a law . 51 7 Fundamental groups of asymptotic cones 53 7.1 Preliminaries on nets . 53 7.2 Construction of the group . 57 7.3 Tree-graded asymptotic cones . 60 7.4 Free products appearing as fundamental groups of asymptotic cones . 66 7.5 Groups with continuously many non-homeomorphic asymptotic cones . 67 8 Asymptotically tree-graded groups are relatively hyperbolic 69 8.1 Weak relative hyperbolicity . 70 8.1.A A characterization of hyperbolicity . 70 8.1.B Generalizations of already proven results and new results . 71 8.1.C Hyperbolicity of Cayley(G; S [H)...................... 73 8.2 The BCP Property . 79 8.3 Undistorted subgroups of relatively hyperbolic groups . 82 9 Appendix. Relatively hyperbolic groups are asymptotically tree-graded. By Denis Osin and Mark Sapir 85 References 88 1 Introduction An asymptotic cone of a metric space is, roughly speaking, what one sees when one looks at the space from in¯nitely far away. More precisely, any asymptotic cone of a metric space (X; dist) corresponds to an ultra¯lter !, a sequence of observation points e = (en)n2N from X and a sequence of scaling constants d = (dn)n2N diverging to +1. The cone Con(X; e; d) corresponding to e and d is the !-limit of the sequence of spaces with basepoints (X; dist=dn; en) (see Section 3 for precise de¯nitions). In particular, if X is the Cayley graph of a group G with a word metric then the asymptotic cones of X are called asymptotic cones of G. The concept of asymptotic cone was essentially used by Gromov in [Gr1] and then formally introduced by van den Dries and Wilkie [VDW]. Asymptotic cones have been used to characterize important classes of groups: ² A ¯nitely generated group is virtually Abelian if and only if its asymptotic cones are n isometric to the Euclidean space R ([Gr1], [Pa]). ² A ¯nitely generated group is virtually nilpotent if and only if its asymptotic cones are locally compact ([Gr1], [VDW], [Dr4]). ² A ¯nitely generated group is hyperbolic if and only if all its asymptotic cones are R-trees ([Gr3]). In [DP1] it is shown moreover that asymptotic cones of non-elementary hyperbolic groups are all isometric to the complete homogeneous R-tree of valence continuum. The asymptotic 2 cones of elementary groups are isometric to either a line R (if the group is in¯nite) or to a point. In particular, every hyperbolic group has only one asymptotic cone up to isometry. Asymptotic cones of quasi-isometric spaces are bi-Lipschitz equivalent. In particular the topology of an asymptotic cone of a ¯nitely generated group does not depend on the choice of the generating set. This was used in [KaL1] and [KaL2] to prove rigidity results for fundamental groups of Haken manifolds, in [KlL] to prove rigidity for cocompact lattices in higher rank semisimple groups, and in [Dr2] to provide an alternative proof of the rigidity for non-cocompact lattices in higher rank semisimple groups. For a survey of results on quasi-isometry invariants and their relations to asymptotic cones see [Dr4]. The power of asymptotic cones stems from the fact that they capture both geometric and logical properties of the group, since a large subgroup of the ultrapower ¦G=! of the group G acts transitively by isometries on the asymptotic cone Con!(G; e; d). Logical aspects of asymptotic cones are studied and used in the recent papers by Kramer, Shelah, Tent and Thomas [KSTT], [KT]. One of the main properties of asymptotic cones of a metric space X is that geometry of ¯nite con¯gurations of points in the asymptotic cone reflects the \coarse" geometry of similar ¯nite con¯gurations in X. This is the spirit of Gromov-Delzant's approximation statement [CDP] and of the applications of R-trees to Rips-Sela theory of equations in hyperbolic groups and homomorphisms of hyperbolic groups [RiSe]. This was also used in Drut»u's proof of hyperbolicity of groups with sub-quadratic isoperimetric inequality [Dr3]. By a result of Gromov [Gr3] if all asymptotic cones of a ¯nitely presented group are simply connected then the group has polynomial isoperimetric function and linear isodiametric function. Papasoglu proved in [Pp] that groups having quadratic isoperimetric functions have simply connected asymptotic cones. In general, asymptotic cones of groups are not necessarily simply connected [Tr]. In fact, if a group G is not ¯nitely presented then its asymptotic cones cannot all be simply connected [Gr3, Dr4]. A higher-dimensional version of this result is obtained by Riley [Ri]. According to the result of Gromov cited above, examples of ¯nitely presented groups with non-simply connected asymptotic cones can be found in [Bri] and [SBR]. Although asymptotic cones can be completely described in some cases, the general perception is nevertheless that asymptotic cones are usually large and \undescribable". This might be the reason of uncharacteristically \mild" questions by Gromov [Gr3]: Problem 1.1. Which groups can appear as subgroups in fundamental groups of asymptotic cones of ¯nitely generated groups? Problem 1.2. Is it true that the fundamental group of an asymptotic cone of a group is either trivial or uncountable? In [Gr3], Gromov also asked the following question. Problem 1.3. How many non-isometric asymptotic cones can a ¯nitely generated group have? A solution of Problem 1.1 was given by Erschler and Osin [EO]. They proved that every metric space satisfying some weak properties can be ¼1- and isometrically embedded into the asymptotic cone of a ¯nitely generated group. This implies that every countable group is a subgroup of the fundamental group of an asymptotic cone of a ¯nitely generated group. Notice that since asymptotic cones tend to have fundamental groups of order continuum, this result does not give information about the structure of the whole fundamental group of an asymptotic cone, or about how large the class of di®erent asymptotic cones is: there exists a group of cardinality continuum that contains all countable groups as subgroups (for example, 3 the group of all permutations of a countable set). One of the goals of this paper is to get more precise information about fundamental groups of asymptotic cones, and about the whole set of di®erent asymptotic cones of a ¯nitely generated group. Problem 1.3 turned out to be related to the Continuum Hypothesis (i.e. the famous question @ of whether there exists a set of cardinality strictly between @0 and 2 0 ). Namely, in [KSTT], it is 2@0 proved that if the Continuum Hypothesis is not true then any uniform lattice in SLn(R) has 2 non-isometric asymptotic cones, but if the Continuum Hypothesis is true then any uniform lattice in SLn(R) has exactly one asymptotic cone up to isometry, moreover the maximal theoretically possible number of non-isometric asymptotic cones of a ¯nitely generated group is continuum. Recall that the Continuum Hypothesis is independent of the usual axioms of set theory (ZFC). It is known, however, that even if the Continuum Hypothesis is true, there exist groups with more than one non-homeomorphic asymptotic cones [TV]. Nevertheless, it was not known whether there exists a group with the maximal theoretically possible number of non-isometric asymptotic cones (continuum). In [Gr2], Gromov introduced a useful generalization of hyperbolic groups, namely the rela- tively hyperbolic groups1. This class includes: (1) geometrically ¯nite Kleinian groups; these groups are hyperbolic relative to their cusp subgroups; (2) fundamental groups
Load more

Recommended publications
  • March, 2016 ALEXANDER YU. OLSHANSKIY Address
    Curriculum Vitae - March, 2016 ALEXANDER YU. OLSHANSKIY Address: Department of Mathematics Vanderbilt University 1326 Stevenson Ctr Nashville, TN 37240 Electronic address: [email protected] EDUCATION AND DEGREES: Undergraduate: Moscow State University (1963-1968); Graduate: Moscow State University (1968-1970); B.S. in Mathematics, 1968; Ph.D. in Mathematics, 1971; Doctor of Science in Mathematics (Habilitation), 1979. POSITIONS HELD: Assistant Professor of Mathematics, Moscow State University, 1970-1977 Associate Professor of Mathematics, Moscow State University, 1978-1984 Professor of Mathematics, Moscow State University, 1985- Centennial Professor of Mathematics, Vanderbilt University, 1999-present AWARDS, GRANTS AND HONORS: Moscow Mathematical Society Prize winner, 1970 Invited speaker of International Congress of Mathematics, 1982 Malcev prize of the Russian Academy of Sciences for the series of papers in geometric and combinatorial group theory, 2000 Fellow of the American Mathematical Society, class 2015. Kargapolov Prize "for outstanding contributions to group theory" at the 3rd International Algebra Conference, Krasnoyarsk, Russia, 1993 Honorary Badge, Seminar "Algebra and Logic," Novosibirsk. G.A. Miller Visiting Professor, University of Illinois at Urbana Cham- paign, March-April 1995 Russian Federation Presidential Award 1993-1996 INTAS Award 1994-1998 Soros Professorship, 1995-96 and 1998 Russian Fund (awarded to outstanding research groups), 1996-1997 INTAS grant 99-1224, 2000-2003 Russian Fund for Fundamental Research, 1996-1998, 1999-2001; grant 02-01-00170, 2002-2004; grant 05-01-00895, 2005-2007; grant 08-01-00573, 2008-2010; grant 11-01-00945, 2011-2013; grant 15-01-05823, 2015-2017. 1 NSF Focus Research Grant, DMS-0455881, 2005-2008. NSF research grants (with Mark Sapir): DMS 0072307, 2000- 2003; DMS-0245600, 2003-2006; DMS 0700811, 2007 - 2012; DMS 1161294, 2012- 2015, DMS-1500180, 2015-2018.
    [Show full text]
  • Morse, Contracting, and Strongly Contracting Sets with Applications to Boundaries and Growth of Groups
    Morse, contracting, and strongly contracting sets with applications to boundaries and growth of groups Christopher H. Cashen, PhD Faculty of Mathematics University of Vienna 1090 Vienna,Austria E-mail address: [email protected] URL: http://www.mat.univie.ac.at/~cashen Key words and phrases. Morse, contracting, strongly contracting, graphical small cancellation, group action, boundary, growth tightness, cogrowth The author was supported by the Austrian Science Fund (FWF): P 30487-N35 during the writing of this thesis. Support for the individual papers can be found in the respective chapters. Abstract. We investigate several quantitative generalizations of the notion of quasiconvex subsets of (Gromov) hyperbolic spaces to arbitrary geodesic metric spaces. Some of these, such as the Morse property, strong contraction, and superlinear divergence, had been studied before in more specialized contexts, and some, such as contraction, we introduce for the first time. In general, we prove that quasiconvexity is the weakest of the properties, strong contraction is the strongest, and all of the others are equivalent. However, in hyperbolic spaces all are equivalent, and we prove that in CAT(0) spaces all except quasiconvexity are equivalent. Despite the fact that many of these properties are equivalent, they are useful for different purposes. For instance, it is easy to see that the Morse property is quasiisometry invariant, but the contraction property gives good control over the divagation behavior of geodesic rays with a common basepoint. We exploit this control to define a boundary for arbitrary finitely generated groups that shares some properties of the boundary of a hyperbolic group.
    [Show full text]
  • Combinatorial and Geometric Group Theory
    Combinatorial and Geometric Group Theory Vanderbilt University Nashville, TN, USA May 5–10, 2006 Contents V. A. Artamonov . 1 Goulnara N. Arzhantseva . 1 Varujan Atabekian . 2 Yuri Bahturin . 2 Angela Barnhill . 2 Gilbert Baumslag . 3 Jason Behrstock . 3 Igor Belegradek . 3 Collin Bleak . 4 Alexander Borisov . 4 Lewis Bowen . 5 Nikolay Brodskiy . 5 Kai-Uwe Bux . 5 Ruth Charney . 6 Yves de Cornulier . 7 Maciej Czarnecki . 7 Peter John Davidson . 7 Karel Dekimpe . 8 Galina Deryabina . 8 Volker Diekert . 9 Alexander Dranishnikov . 9 Mikhail Ershov . 9 Daniel Farley . 10 Alexander Fel’shtyn . 10 Stefan Forcey . 11 Max Forester . 11 Koji Fujiwara . 12 Rostislav Grigorchuk . 12 Victor Guba . 12 Dan Guralnik . 13 Jose Higes . 13 Sergei Ivanov . 14 Arye Juhasz . 14 Michael Kapovich . 14 Ilya Kazachkov . 15 i Olga Kharlampovich . 15 Anton Klyachko . 15 Alexei Krasilnikov . 16 Leonid Kurdachenko . 16 Yuri Kuzmin . 17 Namhee Kwon . 17 Yuriy Leonov . 18 Rena Levitt . 19 Artem Lopatin . 19 Alex Lubotzky . 19 Alex Lubotzky . 20 Olga Macedonska . 20 Sergey Maksymenko . 20 Keivan Mallahi-Karai . 21 Jason Manning . 21 Luda Markus-Epstein . 21 John Meakin . 22 Alexei Miasnikov . 22 Michael Mihalik . 22 Vahagn H. Mikaelian . 23 Ashot Minasyan . 23 Igor Mineyev . 24 Atish Mitra . 24 Nicolas Monod . 24 Alexey Muranov . 25 Bernhard M¨uhlherr . 25 Volodymyr Nekrashevych . 25 Graham Niblo . 26 Alexander Olshanskii . 26 Denis Osin . 27 Panos Papasoglu . 27 Alexandra Pettet . 27 Boris Plotkin . 28 Eugene Plotkin . 28 John Ratcliffe . 29 Vladimir Remeslennikov . 29 Tim Riley . 29 Nikolay Romanovskiy . 30 Lucas Sabalka . 30 Mark Sapir . 31 Paul E. Schupp . 31 Denis Serbin . 32 Lev Shneerson .
    [Show full text]
  • Addendum Median Structures on Asymptotic Cones and Homomorphisms Into Mapping Class Groups
    e Proc. London Math. Soc. (3) 102 (2011) 555–562 C 2011 London Mathematical Society doi:10.1112/plms/pdr008 Addendum Median structures on asymptotic cones and homomorphisms into mapping class groups (Proc. London Math. Soc. (3) (2011) 503–554) Jason Behrstock, Cornelia Drut¸u and Mark Sapir The goal of this addendum to [1] is to show that our methods together with a result of Bestvina, Bromberg and Fujiwara [3, Proposition 5.9] yield a proof of the following theorem. Theorem 1. If a finitely presented group Γ has infinitely many pairwise non-conjugate homomorphisms into MCG(S), then Γ virtually splits (virtually acts non-trivially on a simplicial tree). This theorem is a particular case of a result announced by Groves.† From private emails received by the authors, it is clear that the methods used by Groves are significantly different. Note that the same new methods allow us to give another proof of the finiteness of the set of homomorphisms from a group with property (T) to a mapping class group [1, Theorem 1.2], which is considerably shorter than our original proof; see Corollary 6 and the discussion following it. Theorem 1.2 in [1] may equally be obtained from Theorem 1 and the fact that every group with property (T) is a quotient of a finitely presented group with property (T) (see [11, Theorem p. 5]). The property of the mapping class groups given in Theorem 1 can be viewed as another ‘rank 1’ feature of these groups. In contrast, note that a recent result of [8] shows that the rank 2 lattice SL3(Z) contains infinitely many pairwise non-conjugate copies of the triangle group Δ(3, 3, 4) = a, b | a3 = b3 =(ab)4 =1.
    [Show full text]
  • Quasi-Isometric Rigidity in Low Dimensional Topology
    Quasi-isometric rigidity in low dimensional topology Jason Behrstock (Lehman College, CUNY), Michael Kapovich (University of California, Davis), Walter Neumann (Columbia University) March 8, 2010–March 12, 2010 1 Overview of the field and recent developments The early work of Mostow, Margulis and Prasad on rigidity of arithmetic lattices has evolved into a broad use of quasi-isometry techniques in group theory and low dimensional topology. The word metric on a finitely generated group makes it into a metric space which is uniquely determined up to the geometric relation called quasi-isometry, despite the fact that the metric depends on the choice of generating set. As for lattices in suitable Lie groups, where quasi-isometry of lattices implies commensurability, the general quasi-isometric study of groups aims to understand the remarkable extent to which this completely geometric notion often captures algebraic properties of the group. The Milnor-Schwarz Lemma provides an equivalence between the geometry of the word metric on the fundamental group of a compact Riemannian manifold (or metric complex) with the geometry of its universal cover. So the quasi-isometry study of groups also returns information about the spaces. This relationship has proved particularly productive in low dimensional geometry/topology. There are currently a large variety of groups whose quasi-isometric geometry is actively being studied by geometric group theorists. Many of these groups have close relations to objects studied by low dimen- sional topologists. Examples of these include automorphism groups of free groups, mapping class groups (Hamenstadt,¨ Behrstock-Kleiner-Minsky-Mosher), 3-manifold groups (Gromov-Sullivan, Cannon-Cooper, Eskin-Fisher-Whyte, Kapovich-Leeb, Rieffel, Schwartz, Behrstock-Neumann), solvable Lie groups (Eskin- Fisher-Whyte, Dymarz; the 3-dimensional group Solv had long been the holdout in understanding geomet- ric 3-manifold groups), Artin groups (Bestvina-Kleiner-Sageev, Behrstock-Neumann), relatively hyperbolic groups (Drutu-Sapir, Osin), and others.
    [Show full text]
  • Lectures on Geometric Group Theory
    Lectures on Geometric Group Theory Cornelia Drutu and Michael Kapovich Preface The main goal of this book is to describe several tools of the quasi-isometric rigidity and to illustrate them by presenting (essentially self-contained) proofs of several fundamental theorems in this area: Gromov’s theorem on groups of polynomial growth, Mostow Rigidity Theorem and Schwartz’s quasi-isometric rigidity theorem for nonuniform lattices in the real-hyperbolic spaces. We conclude with a survey of the quasi-isometric rigidity theory. The main idea of the geometric group theory is to treat finitely-generated groups as geometric objects: With each finitely-generated group G one associates a metric space, the Cayley graph of G. One of the main issues of the geometric group theory is to recover as much as possible algebraic information about G from the geometry of the Cayley graph. (A somewhat broader viewpoint is to say that one studies a finitely generated group G by analyzing geometric properties of spaces X on which G acts geometrically, i.e., properly discontinuously, cocompactly and isometrically. The Cayley graph is just one of such spaces.) A primary obstacle for this is the fact that the Cayley graph depends not only on G but on a particular choice of a generating set of G. Cayley graphs associated with different generating sets are not isometric but quasi-isometric. The fundamental question which we will try to address in this book is: If G; G0 are quasi-isometric groups, to which extent G and G0 share the same algebraic properties? The best one can hope here is to recover the group G up to virtual isomorphism from its geometry.
    [Show full text]
  • Algorithmic and Asymptotic Properties of Groups 3 Letter to the Left
    Algorithmic and asymptotic properties of groups Mark Sapir∗ Abstract. This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Hig- man embeddings, and constructions of finitely presented groups with extreme properties (monsters). Mathematics Subject Classification (2000). Primary 20F65; Secondary 20F10. Keywords. Turing machine, S-machine, Dehn function, Higman embedding, conjugacy problem, amenable group. 1. Introduction Although the theory of infinite groups is very rich and full of powerful results, there are very few results having more influence on group theory and surrounding areas of mathematics (especially geometry and topology) as the following five: • The Boone-Novikov theorem about existence of finitely presented groups with undecidable word problem [8],[35]; • The Higman theorem about embeddability of recursively presented groups into finitely presented groups [28]; • The Adian-Novikov solution of the Burnside problem [36]; • Gromov’s theorem about groups with polynomial growth [23]; • Olshanskii and his students’ theorems about existence of groups with all arXiv:math/0602215v1 [math.GR] 10 Feb 2006 proper subgroups cyclic (Tarski monsters), and other finitely generated groups with extreme properties [38]. In this paper, I am going to survey the last ten years of my work on the topics related to these results. Acknowledgement. Most of the work surveyed here is joint with J.-C. Birget, C. Drut¸u, V. Guba, A. Olshanskii and E. Rips. I am very grateful to them for cooperation. ∗This work was supported in part by the NSF grants DMS 0245600 and DMS-0455881. 2 Mark Sapir 2.
    [Show full text]
  • Young Geometric Group Theory V
    Young Geometric Group Theory V Karlsruhe, Germany Monday, the 15th of February, 2016 – Friday, the 19th of February, 2016 Contents 1 Minicourse speakers 1 2 Other speakers 4 3 Poster session 10 4 Research statements 12 5 List of participants 129 Mon, Feb. 15th Tue, Feb. 16th Wed Feb. 17th Thu, Feb. 18th Fri, Feb. 19th 08:30 - 09:00 registration registration 09:00 - 09:50 Iozzi 1 Mosher 2 Iozzi 2 Vogtmann 3 Iozzi 4 10:00 - 10:50 Mosher 1 Vogtmann 2 Bridson 4 Mosher 4 Vogtmann 4 11:00 - 11:30 coffee and tea coffee and tea coffee and tea coffee and tea coffee and tea 11:30 - 12:20 Bou-Rabee / Mann Dowdall / Meiri Mosher 3 Horbez / Puder Touikan 12:30 - 14:20 lunch lunch lunch lunch lunch and registration 14:30 - 15:20 Vogtmann 1 Bridson 2 j Iozzi 3 15:30 - 16:00 coffee and tea coffee and tea excursion coffee and tea 16:00 - 16:50 Bridson 1 Bridson 3 j Hull / Wade 17:00 - 17:50 tutorials/discussions tutorials/discussions j tutorials/discussions 19:00 - — poster session 1 Minicourse speakers Martin Bridson (University of Oxford, UK) Profinite recognition of groups, Grothendieck Pairs, and low- dimensional orbifolds I shall begin by discussing the history of the following general problem: to what extent is a residually-finite group determined by its set of finite quotients (equivalently, its profinite completion)? A precise instance of this question was posed by Grothendieck in 1970: he asked if there could exist pairs of finitely presented, residually finite groups H < G such that the inclusion map induces an isomorphism of profinite completions but H is not isomorphic to G.
    [Show full text]
  • DIVERGENCE in LATTICES in SEMISIMPLE LIE GROUPS and GRAPHS of GROUPS 3 Example, a Group Has Linear Or Superlinear Divergence Without Specifying a Generating Set
    DIVERGENCE IN LATTICES IN SEMISIMPLE LIE GROUPS AND GRAPHS OF GROUPS CORNELIA DRUT¸U, SHAHAR MOZES AND MARK SAPIR Abstract. Divergence functions of a metric space estimate the length of a path connecting two points A, B at distance ≤ n avoiding a large enough ball around a third point C. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. That property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb- Kaimanovich-Masur which states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. We also deduce a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy. We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has “many” periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditch’s properties that are weaker than local compactness. This gives a new proof of Behrstock’s result that every pseudo-Anosov element in a mapping class group is Morse. On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the Q-rank is 1 and when the lattice is SLn(OS ) where n ≥ 3, S is a finite set of valuations of a number field K including all infinite valuations, and OS is the corresponding ring of S-integers.
    [Show full text]
  • Annualreport 2010 2011
    C CENTRE R DERECHERCHES M MATHÉMATIQUES AnnualReport 2010 2011 . i C ii C CENTRE R DERECHERCHES M MATHÉMATIQUES AnnualReport 2010 2011 . iii Centre de recherches mathématiques Université de Montréal C.P. 6128, succ. Centre-ville Montréal, QC H3C 3J7 Canada [email protected] Also available on the CRM website http://crm.math.ca/docs/docRap_an.shtml. © Centre de recherches mathématiques Université de Montréal, 2012 ISBN 978-2-921120-49-4 C Presenting the Annual Report 2010 – 2011 1 ematic Program 4 ematic Programs of the Year 2010 – 2011: “Geometric, Combinatorial and Computational Group e- ory” and “Statistics” ............................................ 5 Aisenstadt Chairholders in 2010 – 2011: Yuri Gurevich, Angus Macintyre, Alexander Razborov, and James Robins ................................................ 6 Activities of the ematic Semesters ...................................... 9 Past ematic Programs ............................................. 21 General Program 23 CRM activities .................................................. 24 Colloquium Series ................................................ 36 Multidisciplinary and Industrial Program 39 Activities of the Climate Change and Sustainability Program ........................ 40 Activities of the Multidisciplinary and Industrial Program .......................... 41 CRM Prizes 47 CRM – Fields – PIMS Prize 2011 Awarded to Mark Lewis ........................... 48 André-Aisenstadt Prize 2011 Awarded to Joel Kamnitzer ........................... 48 CAP – CRM Prize 2011 Awarded
    [Show full text]
  • 2015 Annual Report
    ANNUAL REPORT 2015 CLAY MATHEMATICS INSTITUTE CMI1 CLAY MATHEMATICS INSTITUTE Mission The primary objectives and purposes of the Clay Mathematics Institute are: n to increase and disseminate mathematical knowledge n to educate mathematicians and other scientists about new discoveries in the field of mathematics n to encourage gifted students to pursue mathematical careers n to recognize extraordinary achievements and advances in mathematical research The CMI will further the beauty, power and universality of mathematical thought. The Clay Mathematics Institute is governed by its Board of Directors, Scientific Advisory Board and President. Board meetings are held to consider nominations and research proposals and to conduct other business. The Scientific Advisory Board is responsible for the approval of all proposals and the selection of all nominees. BOARD OF DIRECTORS AND EXECUTIVE OFFICERS Landon T. Clay, Chairman, Director, Treasurer Lavinia D. Clay, Director, Secretary Thomas Clay, Director Nicholas Woodhouse, President Brian James, Chief Administrative Officer SCIENTIFIC ADVISORY BOARD Simon Donaldson, Imperial College London and Simons Center, Stony Brook University Mike Hopkins, Harvard University Carlos Kenig, University of Chicago Andrei Okounkov, Columbia University Andrew Wiles, University of Oxford OXFORD STAFF Naomi Kraker, Administrative Manager Anne Pearsall, Administrative Assistant AUDITORS Wolf & Company, P.C., 99 High Street, Boston, MA 02110 LEGAL COUNSEL Sullivan & Worcester LLP, One Post Office Square, Boston, MA
    [Show full text]
  • Algorithmic and Asymptotic Properties of Groups
    Algorithmic and asymptotic properties of groups Mark Sapir∗ Abstract. This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters). Mathematics Subject Classification (2000). Primary 20F65; Secondary 20F10. Keywords. Turing machine, S-machine, Dehn function, Higman embedding, conjugacy prob- lem, amenable group. 1. Introduction Although the theory of infinite groups is very rich and full of powerful results, there are very few results having more influence on group theory and surrounding areas of mathematics (especially geometry and topology) as the following five. • The Boone–Novikov theorem about existence of finitely presented groups with undecidable word problem [8], [35]. • The Higman theorem about embeddability of recursively presented groups into finitely presented groups [28]; • The Adian–Novikov solution of the Burnside problem [36]. • Gromov’s theorem about groups with polynomial growth [23]. • Olshanskii and his students’theorems about existence of groups with all proper subgroups cyclic (Tarski monsters), and other finitely generated groups with extreme properties [38]. In this paper, I am going to survey the last ten years of my work on the topics related to these results. Acknowledgement. Most of the work surveyed here is joint with J.-C. Birget, C. Dru¸tu, V. Guba, A. Olshanskii and E. Rips. I am very grateful to them for co- operation. ∗ This work was supported in part by the NSF grants DMS 0245600 and DMS-0455881. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society 224 Mark Sapir 2.
    [Show full text]