The geometry and topology of Coxeter groups

These notes are intended as an introduction to the theory of Coxeter groups. They closely follow my talk in the Lectures on Modern Mathematics Series at the Mathematical Sciences Center in Tsinghua University on May 10, 2013. They were prepared from the beamer presentation which I used during my talk. AMS classification numbers. Primary: 20F55, 20F65

[1]  B. Rémy,et al.  Simplicité abstraite des groupes de Kac–Moody non affines , 2006 .

[2]  S. Eilenberg,et al.  On the Lusternik-Schnirelmann Category of Abstract Groups , 1957 .

[3]  Robert D. Edwards,et al.  The Topology of Manifolds and Cell-Like Maps , 2010 .

[4]  K. S. Brown Homological criteria for finiteness , 1975 .

[5]  Ruth Charney STRICT HYPERBOLIZATION , 2003 .

[6]  Robert Schrader,et al.  On the curvature of piecewise flat spaces , 1984 .

[7]  John Milnor,et al.  A note on curvature and fundamental group , 1968 .

[8]  A. Reid,et al.  Infinite Coxeter groups are virtually indicarle , 1998, Proceedings of the Edinburgh Mathematical Society.

[9]  Dongwen Qi On Irreducible, Infinite, Non-affine Coxeter Groups , 2006, math/0603670.

[10]  F. Farrell,et al.  On Novikov's Conjecture for Non-Positively Curved Manifolds, I , 1981 .

[11]  H. Busemann Spaces with non-positive curvature , 1948 .

[12]  Igor Nikolaev,et al.  Generalized Riemannian spaces , 1986 .

[13]  R. Varga,et al.  Proof of Theorem 4 , 1983 .

[14]  J. Crisp Symmetrical Subgroups of Artin Groups , 2000 .

[15]  Tadeusz Januszkiewicz,et al.  Cohomology of buildings and of their automorphism groups , 2002 .

[16]  William J. Floyd,et al.  Symmetries of planar growth functions , 1988 .

[17]  Huai-Dong Cao,et al.  A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow , 2006 .

[18]  Ki Hang Kim,et al.  Homology of certain algebras defined by graphs , 1980 .

[19]  Local homology properties of boundaries of groups. , 1996 .

[20]  Ennis Rosas,et al.  RIGIDITY THEOREMS FOR RIGHT ANGLED REFLECTION GROUPS , 1988 .

[21]  E. Witt,et al.  Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe , 1941 .

[22]  Ruth Charney,et al.  The (,1)-problem for hyperplane complements associated to infinite reflection groups , 1995 .

[23]  Jeff Cheeger,et al.  L2-Cohomology and group cohomology , 1986 .

[24]  È. Vinberg,et al.  Some linear groups virtually having a free quotient. , 2000 .

[25]  A. Kaul Rigidity for a class of Coxeter groups , 2000 .

[26]  Geodesics in piecewise linear manifolds , 1976 .

[27]  N. Meyers,et al.  H = W. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[28]  J. W. Cannon,et al.  Shrinking cell-like decompositions of manifolds. Codimension three , 1979 .

[29]  John McCleary,et al.  User's Guide to Spectral Sequences , 1985 .

[30]  Gabor Moussong,et al.  Hyperbolic Coxeter groups , 1988 .

[31]  H. Minkowski,et al.  Zur Theorie der positiven quadratischen Formen. , 1887 .

[32]  T. Januszkiewicz,et al.  Hyperbolization of polyhedra , 1991 .

[33]  R. Lashof,et al.  Problems in Differential and Algebraic Topology. Seattle Conference, 1963 , 1965 .

[34]  Homological algebra of Novikov-Shubin invariants and morse inequalities , 1996, dg-ga/9606013.

[35]  William P. Thurston,et al.  Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds , 1986, math/9801019.

[36]  Michael Davis Some aspherical manifolds , 1987 .

[37]  B. Nucinkis,et al.  Some groups of type VF , 2003 .

[38]  Luis Paris Irreducible Coxeter Groups , 2007, Int. J. Algebra Comput..

[39]  G. Lusztig Left cells in weyl groups , 1983 .

[40]  Jacques Tits,et al.  A Local Approach to Buildings , 1981 .

[41]  Daniel E Cohen,et al.  Groups of Cohomological Dimension One , 1972 .

[42]  S. Weinberger The Topological Classification of Stratified Spaces , 1995 .

[43]  F. Farrell Poincaré daulity and groups of type (FP) , 1975 .

[44]  Allan M. Sinclair,et al.  The basic construction , 2008 .

[45]  The FAn Conjecture for Coxeter groups , 2005, math/0509439.

[46]  Michael W. Davis The cohomology of a Coxeter group with group ring coefficients , 1998 .

[47]  R. Ho Algebraic Topology , 2022 .

[48]  Friedrich Hirzebruch,et al.  Groupes et géométries de Coxeter , 2001 .

[49]  F. Raymond,et al.  Separation and union theorems for generalized manifolds with boundary. , 1960 .

[50]  G. Perelman The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[51]  Nellie Clarke Brown Trees , 1896, Savage Dreams.

[52]  È. Vinberg,et al.  Strong Tits alternative for subgroups of Coxeter groups. , 2002 .

[53]  P. A. Smith Transformations of Finite Period. II , 1938 .

[54]  Curtis T. McMullen,et al.  Coxeter groups, Salem numbers and the Hilbert metric , 2002 .

[55]  J. Hausmann,et al.  Aspherical manifolds without smooth or PL structure , 1989 .

[56]  M. Kreck The Novikov Conjecture: Geometry and Algebra , 2005 .

[57]  Swiatoslaw R. Gal Real Root Conjecture Fails for Five- and Higher-Dimensional Spheres , 2005, Discret. Comput. Geom..

[58]  F. Raymond,et al.  Manifolds covered by Euclidean space , 1975 .

[59]  É. Ghys,et al.  Sur Les Groupes Hyperboliques D'Apres Mikhael Gromov , 1990 .

[60]  E. Robertson,et al.  Buildings and group amalgamations , 1987 .

[61]  Walter Feit,et al.  The nonexistence of certain generalized polygons , 1964 .

[62]  Mladen Bestvina,et al.  The boundary of negatively curved groups , 1991 .

[63]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .

[64]  John Stallings,et al.  Group Theory and Three-dimensional Manifolds , 1971 .

[65]  E C G Sudarshan,et al.  Lie Groups and Lie Algebras , 2010 .

[66]  H. Hopf Fundamentalgruppe und zweite Bettische Gruppe , 1941 .

[67]  R. Strebel A remark on subgroups of infinite index in Poincaré duality groups , 1977 .

[68]  F. Farrell,et al.  A topological analogue of Mostow’s rigidity theorem , 1989 .

[69]  R. G. Möller Groups acting on graphs , 1991 .

[70]  Jacek Świa¸tkowski Trivalent Polygonal Complexes of Nonpositive Curvature and Platonic Symmetry , 1998 .

[71]  Mark Ronan,et al.  Lectures on Buildings , 1989 .

[72]  Ruth Charney,et al.  Singular Metrics of Nonpositive Curvature on Branched Covers of Riemannian Manifolds , 1993 .

[73]  M. Newman,et al.  The Engulfing Theorem for Topological Manifolds , 1966 .

[74]  Warren Dicks,et al.  Presentations for subgroups of Artin groups , 2012, 1210.6585.

[75]  John Stallings,et al.  Polyhedral homotopy-spheres , 1960 .

[76]  R. Kirby,et al.  On the triangulation of manifolds and the Hauptvermutung , 1969 .

[77]  B. Mühlherr The isomorphism problem for Coxeter groups , 2005 .

[78]  Ruth Charney,et al.  Finite K (π, 1)s for Artin Groups , 1996 .

[79]  M. Atiyah Elliptic operators, discrete groups and von Neumann algebras , 1976 .

[80]  J. Munkres Obstructions to imposing differentiable structures , 1964 .

[81]  Finite automorphisms of negatively curved Poincaré duality groups , 2003, math/0302218.

[82]  J. Whitehead Certain theorems about three-dimensional manifolds (i) , 1934 .

[83]  D. L. Johnson Topics in the Theory of Group Presentations: Small cancellation groups , 1980 .

[84]  W. Browder Surgery on Simply-Connected Manifolds , 1972 .

[85]  B. Eckmann Poincaré Duality Groups of Dimension Two are Surface Groups , 1987 .

[86]  B. Eckmann,et al.  Poincaré duality groups of dimension two , 1980 .

[87]  Vinay V. Deodhar A note on subgroups generated by reflections in Coxeter groups , 1989 .

[88]  I. Kaplinskaya Discrete groups generated by reflections in the faces of symplicial prisms in Lobachevskian spaces , 1974 .

[89]  M. Bridson,et al.  Complexes of Groups , 1999 .

[90]  Ruth Charney,et al.  Notes on Locally CAT(l) Spaces , 1995 .

[91]  F. Farrell,et al.  Topological rigidity for compact non-positively curved manifolds , 1990 .

[92]  M. Spivak A comprehensive introduction to differential geometry , 1979 .

[93]  J. Lott,et al.  L 2-Topological Invariants of 3-manifolds by , 1995 .

[94]  Tadeusz Januszkiewicz,et al.  Nonpositive curvature of blow-ups , 1998 .

[95]  G. Perelman Finite extinction time for the solutions to the Ricci flow on certain three-manifolds , 2003, math/0307245.

[96]  J. Whitehead A CERTAIN OPEN MANIFOLD WHOSE GROUP IS UNITY , 1935 .

[97]  È. Vinberg,et al.  DISCRETE LINEAR GROUPS GENERATED BY REFLECTIONS , 1971 .

[98]  Charles F. Miller Decision Problems for Groups — Survey and Reflections , 1992 .

[99]  D. Sullivan HYPERBOLIC GEOMETRY AND HOMEOMORPHISMS , 1979 .

[100]  Ruth Charney,et al.  The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold. , 1995 .

[101]  R. Geoghegan,et al.  The fundamental group at infinity , 1996 .

[102]  M. Newman,et al.  Boundaries of ULC Sets in Euclidean n-Space. , 1948, Proceedings of the National Academy of Sciences of the United States of America.

[103]  G. Perelman Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[104]  M. Gromov,et al.  Hyperbolic Manifolds, Groups and Actions , 1981 .

[105]  Thin buildings , 2005, math/0601005.

[106]  Jacques Tits,et al.  Buildings of Spherical Type and Finite BN-Pairs , 1974 .

[107]  Vanishing theorems and conjectures for the, $\ell ^2$--homology of right-angled Coxeter groups , 2001, math/0102104.

[108]  B. Jackson End invariants of group extensions , 1982 .

[109]  M. Bourdon Immeubles hyperboliques, dimension conforme et rigidité de Mostow , 1997 .

[110]  Bijin Hu Whitehead groups of finite polyhedra with nonpositive curvature , 1993 .

[111]  B. Eckmann Introduction to ℓ2-methods in topology: Reduced ℓ2-homology, harmonic chains, ℓ2-betti numbers , 2000 .

[112]  N. Brady,et al.  On Algebraic and Geometric Dimensions for Groups with Torsion , 2001 .

[113]  M. J. Dunwoody The accessibility of finitely presented groups , 1985 .

[114]  W. Ballmann,et al.  Polygonal complexes and combinatorial group theory , 1994 .

[115]  J. Meier,et al.  The topology at infinity of Coxeter groups and buildings , 2002 .

[116]  W. Thurston Three dimensional manifolds, Kleinian groups and hyperbolic geometry , 1982 .

[117]  R. Howlett,et al.  Automorphisms of nearly finite Coxeter groups. , 2003 .

[118]  W. Lück L 2 -Invariants of Regular Coverings of Compact Manifolds and CW-Complexes , 2001 .

[119]  M. Gromov Kähler hyperbolicity and $L_2$-Hodge theory , 1991 .

[120]  J. Cannon The combinatorial structure of cocompact discrete hyperbolic groups , 1984 .

[121]  László Lovász,et al.  The Shapes of Polyhedra , 1990, Math. Oper. Res..

[122]  H. Pajot,et al.  Rigidity of quasi-isometries for some hyperbolic buildings , 2000 .

[123]  M. Bourdon Sur les immeubles fuchsiens et leur type de quasi-isométrie , 2000, Ergodic Theory and Dynamical Systems.

[124]  Michah Sageev,et al.  Ends of Group Pairs and Non‐Positively Curved Cube Complexes , 1995 .

[125]  On subgroups of Coxeter groups , 2007, 0711.5011.

[126]  The Poincaré Conjecture 99 Years Later : A Progress Report , 2022 .

[127]  G. Dirac On rigid circuit graphs , 1961 .

[128]  Ravi S. Kulkarni,et al.  Regular tessellations of surfaces and (p, q, 2)-triangle groups , 1982 .

[129]  Colin Rourke,et al.  Block bundles: I , 1968 .

[130]  Michael H. Freedman,et al.  The topology of four-dimensional manifolds , 1982 .

[131]  Topology of homology manifolds , 1993, math/9304210.

[132]  T. Dieck Orbittypen und äquivariante Homologie. I , 1972 .

[133]  Marvin Mudrick,et al.  Here They Are , 1972 .

[134]  Michael W. Davis Groups Generated by reflections and aspherical manifolds not covered by Euclidean space , 1983 .

[135]  John Stallings,et al.  On Torsion-Free Groups with Infinitely Many Ends , 1968 .

[136]  A. Lubotzky Free quotients and the first betti number of some hyperbolic manifolds , 1996 .

[137]  Folke Lannér On complexes with transitive groups of automorphisms , 1950 .

[138]  Michael W. Davis,et al.  NOTES ON NONPOSITIVELY CURVED POLYHEDRA , 2004 .

[139]  A. Blumberg BASIC TOPOLOGY , 2002 .

[140]  Michael W. Davis,et al.  When is a Coxeter System Determined by its Coxeter Group? , 2000 .

[141]  D. Kazhdan,et al.  Representations of Coxeter groups and Hecke algebras , 1979 .

[142]  R. Geroch Positive sectional curvatures does not imply positive Gauss-Bonnet integrand , 1976 .

[143]  H. van der Lek,et al.  The homotopy type of complex hyperplane complements , 1983 .

[144]  Constantin Gonciulea Infinite Coxeter groups virtually surject onto Z , 1997 .

[145]  T. Januszkiewicz,et al.  Convex polytopes, Coxeter orbifolds and torus actions , 1991 .

[146]  J. Meier,et al.  When is the graph product of hyperbolic groups hyperbolic? , 1996 .

[147]  J. Tits Structures et groupes de Weyl , 1966 .

[148]  V. Jones Von Neumann Algebras , 2020, Lectures on von Neumann Algebras.

[149]  J. Dodziuk De Rham-Hodge theory for L2-cohomology of infinite coverings , 1977 .

[150]  Bruna Tanaka Cremonini,et al.  Buildings , 1995, Data, Statistics, and Useful Numbers for Environmental Sustainability.

[151]  Daniel T. Wise,et al.  Special Cube Complexes , 2008, The Structure of Groups with a Quasiconvex Hierarchy.

[152]  M. Kervaire Smooth homology spheres and their fundamental groups , 1969 .

[153]  A. Kuku,et al.  Higher Algebraic K-Theory , 2006 .

[154]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[155]  Bernhard Mühlherr,et al.  Rigidity of Coxeter Groups and Artin Groups , 2002 .

[156]  Michael W. Davis Nonpositive Curvature and Reflection Groups , 2001 .

[157]  P. Caprace Conjugacy of 2-spherical subgroups of Coxeter groups and parallel walls , 2005, math/0508057.

[158]  J. Millson On the first Betti number of a constant negatively curved manifold , 1976 .

[159]  C. Wall Finiteness conditions for CW-complex I , 1965 .

[160]  H. Hopf,et al.  Enden offener Räume und unendliche diskontinuierliche Gruppen , 1943 .

[161]  D. Allcock Infinitely many hyperbolic Coxeter groups through dimension 19 , 2006, 0903.0138.

[162]  J. Dodziuk L^2-harmonic forms on rotationally symmetric Riemannian manifolds , 1979 .

[163]  Some examples of discrete group actions on aspherical manifolds , 2003 .

[164]  H. Fischer Boundaries of right-angled Coxeter groups with manifold nerves , 2003 .

[165]  J. Tits,et al.  On buildings and their applications , 1975 .

[166]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1 , 1997, The Mathematical Gazette.

[167]  R. Bott An application of the Morse theory to the topology of Lie-groups , 1956 .

[168]  Richard M. Weiss,et al.  The Structure of Spherical Buildings , 2004 .

[169]  Noel Brady,et al.  Morse theory and finiteness properties of groups , 1997 .

[170]  T. Januszkiewicz,et al.  Fundamental Groups of Blow-ups , 2002 .

[171]  J. Świa̧tkowski,et al.  Hyperbolic Coxeter groups of large dimension , 2003 .

[172]  P. McMullen,et al.  Combinatorially regular polytopes , 1967 .

[173]  Matthew Dyer,et al.  Reflection subgroups of Coxeter systems , 1990 .

[174]  John R. Harper,et al.  Algebraic topology : a first course , 1982 .

[175]  Vinay V. Deodhar On the root system of a coxeter group , 1982 .

[176]  Graham A. Niblo,et al.  Coxeter Groups act on CAT(0) cube complexes , 2003 .

[177]  D. Barnette A proof of the lower bound conjecture for convex polytopes. , 1973 .

[178]  H. Coxeter REGULAR HONEYCOMBS IN HYPERBOLIC SPACE , 1956 .

[179]  Louis Solomon,et al.  A decomposition of the group algebra of a finite Coxeter group , 1968 .

[180]  Tadeusz Januszkiewicz,et al.  Right-angled Artin groups are commensurable with right-angled Coxeter groups , 2000 .

[181]  Robert Steinberg,et al.  Endomorphisms of linear algebraic groups , 1968 .

[182]  Kenneth S. Brown,et al.  Cohomology of Groups , 1982 .

[183]  Bernhard Mühlherr,et al.  On Isomorphisms between Coxeter Groups , 2000, Des. Codes Cryptogr..

[184]  John Milnor,et al.  Towards the Poincaré Conjecture and the Classification of 3-Manifolds, Volume 50, Number 10 , 2003 .

[185]  C. Hodgson,et al.  A characterization of compact convex polyhedra in hyperbolic 3-space , 1993 .

[186]  Daan Krammer,et al.  The conjugacy problem for Coxeter groups , 2009 .

[187]  Michael W. Davis Coxeter groups and aspherical manifolds , 1984 .

[188]  Damian Osajda,et al.  Boundaries of right-angled hyperbolic buildings , 2006, math/0610114.

[189]  W. Jakobsche Homogeneous cohomology manifolds which are inverse limits , 1991 .

[190]  Jacques Tits,et al.  Le problème des mots dans les groupes de Coxeter , 1969 .

[191]  G. Mess Examples of Poincaré duality groups , 1990 .

[192]  Robion Kirby,et al.  Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. , 1977 .

[193]  W. Parry Growth Series of Coxeter Groups and Salem Numbers , 1993 .

[194]  Jean-Pierre Serre Cohomologie des groupes discrets , 1971 .

[195]  C D Papakyriakopoulos,et al.  ON DEHN'S LEMMA AND THE ASPHERICITY OF KNOTS. , 1957, Proceedings of the National Academy of Sciences of the United States of America.

[196]  A. O. Houcine On hyperbolic groups , 2006 .

[197]  W. Lück L2-Betti numbers of mapping tori and groups , 1994 .

[198]  Bruce Kleiner,et al.  Notes on Perelman's papers , 2006 .

[199]  S. Smale Generalized Poincare's Conjecture in Dimensions Greater Than Four , 1961 .

[200]  D. V. Alekseevskij,et al.  Geometry of Spaces of Constant Curvature , 1993 .

[201]  Constantin Gonciulea Virtual epimorphisms of coxeter groups onto free groups , 2000 .

[202]  E. M. Andreev ON CONVEX POLYHEDRA OF FINITE VOLUME IN LOBAČEVSKIĬ SPACE , 1970 .

[203]  E. M. Andreev ON CONVEX POLYHEDRA IN LOBAČEVSKIĬ SPACES , 1970 .

[204]  G. Mostow Strong Rigidity of Locally Symmetric Spaces. , 1973 .

[205]  Abe Shenitzer,et al.  The Poincaré Conjecture? , 2006, Am. Math. Mon..

[206]  R. Howlett Normalizers of Parabolic Subgroups of Reflection Groups , 1980 .

[207]  È. Vinberg,et al.  Spaces of constant curvature , 1993 .

[208]  Michaela Davis The homology of a space on which a reflection group acts , 1987 .

[209]  M. Newman A THEOREM ON PERIODIC TRANSFORMATIONS OF SPACES , 1931 .

[210]  W. Mio Homology manifolds , 1996 .

[211]  Barry Mazur,et al.  A Note on Some Contractible 4-Manifolds , 1961 .

[212]  P. Deligne,et al.  Les immeubles des groupes de tresses généralisés , 1972 .

[213]  P. Ontaneda The double of a hyperbolic manifold and non-positively curved exotic PL structures , 2002 .

[214]  Stratos Prassidis,et al.  Rigidity of Coxeter groups , 2000 .

[215]  J. Stallings The Piecewise-Linear Structure of Euclidean Space , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[216]  H. Freudenthal,et al.  Über die Enden topologischer Räume und Gruppen , 1931 .

[217]  M. Spivak,et al.  Spaces satisfying Poincaré duality , 1967 .

[218]  Y. Benoist A survey on divisible convex sets , 2022 .

[219]  S. Yau,et al.  Compact manifolds of nonpositive curvature , 1972 .

[220]  P. Bahls,et al.  Reflection Independence in Even Coxeter Groups , 2005 .

[221]  H. Donnelly,et al.  On the Differential Form Spectrum of Negatively Curved Riemannian Manifolds , 1984 .

[222]  È. Vinberg,et al.  Hyperbolic reflection groups , 1985 .

[223]  Mathématiques DE L’I.H.É.S,et al.  Volume and bounded cohomology , 2003 .