Skip to main content Accessibility help
×
Hostname: page-component-788cddb947-w95db Total loading time: 0 Render date: 2024-10-08T00:59:22.202Z Has data issue: false hasContentIssue false

Finite and Infinite Quotients of Discrete and Indiscrete Groups

Published online by Cambridge University Press:  15 April 2019

C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
Get access

Summary

These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that construction are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis’ proof of Kneser’s simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agol, Ian, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087, With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553Google Scholar
Agol, Ian, Groves, Daniel, and Manning, Jason Fox, Residual finiteness, QCERF and fillings of hyperbolic groups, Geom. Topol. 13 (2009), no. 2, 1043–1073. MR 2470970CrossRefGoogle Scholar
Agol, Ian, Groves, Daniel, and Manning, Jason Fox, An alternate proof of Wise’s malnormal special quotient theorem, Forum Math. Pi 4 (2016), e1, 54 pp. MR 3456181CrossRefGoogle Scholar
Aschenbrenner, Matthias, Friedl, Stefan, and Wilton, Henry, 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2015. MR 3444187CrossRefGoogle Scholar
Bader, Uri, Caprace, Pierre-Emmanuel, Gelander, Tsachik, and Mozes, Shahar, Simple groups without lattices, Bull. Lond. Math. Soc. 44 (2012), no. 1, 55–67. MR 2881324CrossRefGoogle Scholar
Bader, Uri, Caprace, Pierre-Emmanuel, and Lécureux, Jean, On the linearity of lattices in affine buildings and ergodicity of the singular cartan flow, Preprint, arXiv: 1608.06265, 2016.Google Scholar
Bader, Uri and Shalom, Yehuda, Factor and normal subgroup theorems for lattices in products of groups, Invent. Math. 163 (2006), no. 2, 415–454. MR 2207022CrossRefGoogle Scholar
Bass, Hyman and Lubotzky, Alexander, Tree lattices, Progress in Mathematics, vol. 176, Birkhäuser Boston, Inc., Boston, MA, 2001, With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and Tits, J.. MR 1794898CrossRefGoogle Scholar
Baumslag, Gilbert, A non-cyclic one-relator group all of whose finite quotients are cyclic, J. Austral. Math. Soc. 10 (1969), 497–498. MR 0254127CrossRefGoogle Scholar
Bekka, Bachir, Harpe, Pierre de la, and Valette, Alain, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834CrossRefGoogle Scholar
Belolipetsky, Mikhail, Counting maximal arithmetic subgroups, Duke Math. J. 140 (2007), no. 1, 1–33, With an appendix by Jordan Ellenberg and Akshay Venkatesh. MR 2355066CrossRefGoogle Scholar
Bestvina, Mladen, Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston’s vision, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 53–70. MR 3119822Google Scholar
Bhattacharjee, Meenaxi, Constructing finitely presented infinite nearly simple groups, Comm. Algebra 22 (1994), no. 11, 4561–4589. MR 1284345CrossRefGoogle Scholar
Bleak, Collin and Quick, Martyn, The infinite simple group V of Richard J. Thompson: presentations by permutations, Groups Geom. Dyn. 11 (2017), no. 4, 1401–1436.CrossRefGoogle Scholar
Bondarenko, Ievgen, D’Angeli, Daniele, and Rodaro, Emanuele, The lamplighter group Z3 ʅ Z generated by a bireversible automaton, Comm. Algebra 44 (2016), no. 12, 5257–5268. MR 3520274CrossRefGoogle Scholar
Bondarenko, Ievgen and Kivva, Bohdan, Automaton groups and complete square complexes, Preprint arXiv:1707.00215.Google Scholar
Bridson, Martin R. and Haefliger, André, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486CrossRefGoogle Scholar
Brown, Kenneth S., The geometry of finitely presented infinite simple groups, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 121–136. MR 1230631CrossRefGoogle Scholar
Burger, M., Gelander, T., Lubotzky, A., and Mozes, S., Counting hyperbolic manifolds, Geom. Funct. Anal. 12 (2002), no. 6, 1161–1173. MR 1952926CrossRefGoogle Scholar
Burger, Marc and Mozes, Shahar, Finitely presented simple groups and products of trees, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 7, 747–752. MR 1446574CrossRefGoogle Scholar
Burger, Marc and Mozes, Shahar, Groups acting on trees: from local to global structure, Inst. Hautes Études Sci. Publ. Math. (2000), no. 92, 113–150 (2001). MR 1839488Google Scholar
Burger, Marc and Mozes, Shahar, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. (2000), no. 92, 151–194 (2001). MR 1839489Google Scholar
Burger, Marc, Mozes, Shahar, and Zimmer, Robert J., Linear representations and arithmeticity of lattices in products of trees, Essays in geometric group theory, Ramanujan Math. Soc. Lect. Notes Ser., vol. 9, Ramanujan Math. Soc., Mysore, 2009, pp. 1–25. MR 2605353Google Scholar
Camm, Ruth, Simple free products, J. London Math. Soc. 28 (1953), 66–76. MR 0052420Google Scholar
Cannon, J. W., Floyd, W. J., and Parry, W. R., Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215–256. MR 1426438Google Scholar
Caprace, Pierre-Emmanuel, A sixteen-relator presentation of an infinite hyperbolic Kazhdan group, Preprint arXiv:1708.09772, 2017.Google Scholar
Caprace, Pierre-Emmanuel, Cornulier, Yves, Monod, Nicolas, and Tessera, Romain, Amenable hyperbolic groups, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 11, 2903–2947. MR 3420526CrossRefGoogle Scholar
Caprace, Pierre-Emmanuel and Fujiwara, Koji, Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal. 19 (2010), no. 5, 1296–1319. MR 2585575CrossRefGoogle Scholar
Caprace, Pierre-Emmanuel, Kropholler, Peter H., Reid, Colin D., and Wesolek, Phillip, On the residual and profinite closures of commensurated subgroups, Preprint arXiv: 1706.06853.Google Scholar
Caprace, Pierre-Emmanuel and Boudec, Adrien Le, Bounding the covolume of lattices in products, Preprint arXiv:1805.04469.Google Scholar
Caprace, Pierre-Emmanuel and Monod, Nicolas, Decomposing locally compact groups into simple pieces, Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 1, 97–128. MR 2739075CrossRefGoogle Scholar
Caprace, Pierre-Emmanuel and Monod, Nicolas, A lattice in more than two Kac-Moody groups is arithmetic, Israel J. Math. 190 (2012), 413–444. MR 2956249CrossRefGoogle Scholar
Caprace, Pierre-Emmanuel and Rémy, Bertrand, Simplicity and superrigidity of twin building lattices, Invent. Math. 176 (2009), no. 1, 169–221. MR 2485882CrossRefGoogle Scholar
Caprace, Pierre-Emmanuel and Wesolek, Phillip, Indicability, residual finiteness, and simple subquotients of groups acting on trees, to appear in Geom. Topol.Google Scholar
Champetier, Christophe and Guirardel, Vincent, Limit groups as limits of free groups, Israel J. Math. 146 (2005), 1–75. MR 2151593CrossRefGoogle Scholar
Chinburg, Ted, Friedlander, Holley, Howe, Sean, Kosters, Michiel, Singh, Bhairav, Stover, Matthew, Zhang, Ying, and Ziegler, Paul, Presentations for quaternionic S-unit groups, Exp. Math. 24 (2015), no. 2, 175–182. MR 3350524CrossRefGoogle Scholar
Chinburg, Ted and Stover, Matthew, Small generators for S-unit groups of division algebras, New York J. Math. 20 (2014), 1175–1202. MR 3291615Google Scholar
Coornaert, M., Delzant, T., and Papadopoulos, A., Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups], With an English summary. MR 1075994CrossRefGoogle Scholar
Corlette, Kevin, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135 (1992), no. 1, 165–182. MR 1147961CrossRefGoogle Scholar
Dahmani, F., Guirardel, V., and Osin, D., Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152. MR 3589159Google Scholar
Davidoff, Giuliana, Sarnak, Peter, and Valette, Alain, Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, vol. 55, Cambridge University Press, Cambridge, 2003. MR 1989434Google Scholar
Davis, Michael W., The geometry and topology of Coxeter groups, Introduction to modern mathematics, Adv. Lect. Math. (ALM), vol. 33, Int. Press, Somerville, MA, 2015, pp. 129–142. MR 3445448Google Scholar
Cornulier, Yves de, Guyot, Luc, and Pitsch, Wolfgang, On the isolated points in the space of groups, J. Algebra 307 (2007), no. 1, 254–277. MR 2278053CrossRefGoogle Scholar
Delzant, Thomas, Sous-groupes distingués et quotients des groupes hyperboliques, Duke Math. J. 83 (1996), no. 3, 661–682. MR 1390660CrossRefGoogle Scholar
Deraux, Martin, Parker, John R., and Paupert, Julien, New non-arithmetic complex hyperbolic lattices, Invent. Math. 203 (2016), no. 3, 681–771. MR 3461365CrossRefGoogle Scholar
Dixon, John D., Pyber, László, Seress, Ákos, and Shalev, Aner, Residual properties of free groups and probabilistic methods, J. Reine Angew. Math. 556 (2003), 159–172. MR 1971144Google Scholar
Džambić, Amir and Jones, Gareth A., p-adic Hurwitz groups, J. Algebra 379 (2013), 179–207. MR 3019251CrossRefGoogle Scholar
Gelander, Tsachik and Levit, Arie, Counting commensurability classes of hyperbolic manifolds, Geom. Funct. Anal. 24 (2014), no. 5, 1431–1447. MR 3261631CrossRefGoogle Scholar
Ghys, É. and de la Harpe, P. (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648CrossRefGoogle Scholar
Gilman, Robert, Finite quotients of the automorphism group of a free group, Canad. J. Math. 29 (1977), no. 3, 541–551. MR 0435226CrossRefGoogle Scholar
Glasner, Yair and Mozes, Shahar, Automata and square complexes, Geom. Dedicata 111 (2005), 43–64. MR 2155175CrossRefGoogle Scholar
Grigorchuk, R. I., Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939–985. MR 764305Google Scholar
Grigorchuk, R. I., Just infinite branch groups, New horizons in pro-p groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 121–179. MR 1765119Google Scholar
Gromov, M., Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–213. MR 624814CrossRefGoogle Scholar
Gromov, Mikhael, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981), no. 53, 53–73. MR 623534Google Scholar
Gromov, M., Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829Google Scholar
Gromov, Mikhail and Schoen, Richard, Harmonic maps into singular spaces and padic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. (1992), no. 76, 165–246. MR 1215595Google Scholar
Haglund, Frédéric and Wise, Daniel T., Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551–1620. MR 2377497CrossRefGoogle Scholar
Haglund, Frédéric and Wise, Daniel T., A combination theorem for special cube complexes, Ann. of Math. (2) 176 (2012), no. 3, 1427–1482. MR 2979855CrossRefGoogle Scholar
Higman, Graham, A finitely generated infinite simple group, J. London Math. Soc. 26 (1951), 61–64. MR 0038348Google Scholar
Higman, Graham, Finitely presented infinite simple groups, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974, Notes on Pure Mathematics, No. 8 (1974). MR 0376874Google Scholar
Higman, Graham, Neumann, B. H., and Neumann, Hanna, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254. MR 0032641Google Scholar
Hsu, Tim and Wise, Daniel T., Embedding theorems for non-positively curved polygons of finite groups, J. Pure Appl. Algebra 123 (1998), no. 1-3, 201–221. MR 1492901CrossRefGoogle Scholar
Ivanov, S. V. and Ol’shanskiĬ, A. Yu, Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091–2138. MR 1327257CrossRefGoogle Scholar
Jacobson, Nathan, Basic algebra. II, second ed., W. H. Freeman and Company, New York, 1989. MR 1009787Google Scholar
Janzen, David and Wise, Daniel T., A smallest irreducible lattice in the product of trees, Algebr. Geom. Topol. 9 (2009), no. 4, 2191–2201. MR 2558308CrossRefGoogle Scholar
Kapovich, Ilya and Wise, Daniel T., The equivalence of some residual properties of word-hyperbolic groups, J. Algebra 223 (2000), no. 2, 562–583. MR 1735163CrossRefGoogle Scholar
Katz, Robert A. and Magnus, Wilhelm, Residual properties of free groups, Comm. Pure Appl. Math. 22 (1968), 1–13. MR 0233873Google Scholar
Kimberley, Jason S. and Robertson, Guyan, Groups acting on products of trees, tiling systems and analytic K-theory, New York J. Math. 8 (2002), 111–131. MR 1923572Google Scholar
King, Carlisle S. H., Generation of finite simple groups by an involution and an element of prime order, J. Algebra 478 (2017), 153–173. MR 3621666CrossRefGoogle Scholar
Kneser, Martin, Orthogonale Gruppen über algebraischen Zahlkörpern, J. Reine Angew. Math. 196 (1956), 213–220. MR 0080101Google Scholar
Köhler, Peter, Meixner, Thomas, and Wester, Michael, The affine building of type Ã2 over a local field of characteristic two, Arch. Math. (Basel) 42 (1984), no. 5, 400–407. MR 756691CrossRefGoogle Scholar
Köhler, Peter, Triangle groups, Comm. Algebra 12 (1984), no. 13-14, 1595–1625. MR 743306CrossRefGoogle Scholar
Köhler, Peter, The 2-adic affine building of type Ã2 and its finite projections, J. Combin. Theory Ser. A 38 (1985), no. 2, 203–209. MR 784716CrossRefGoogle Scholar
Kuroš, A. G., Teoriya Grupp, OGIZ, Moscow-Leningrad, 1944. MR 0022843Google Scholar
Boudec, Adrien Le, Groups acting on trees with almost prescribed local action, Comment. Math. Helv. 91 (2016), no. 2, 253–293. MR 3493371CrossRefGoogle Scholar
Liebeck, Martin W., Probabilistic and asymptotic aspects of finite simple groups, Probabilistic group theory, combinatorics, and computing, Lecture Notes in Math., vol. 2070, Springer, London, 2013, pp. 1–34. MR 3026185CrossRefGoogle Scholar
Liebeck, Martin W. and Shalev, Aner, Residual properties of free products of finite groups, J. Algebra 268 (2003), no. 1, 286–289. MR 2005288Google Scholar
Liebeck, Martin W. and Shalev, Aner, Residual properties of the modular group and other free products, J. Algebra 268 (2003), no. 1, 264–285. MR 2005287Google Scholar
Long, D. D. and Reid, A. W., Simple quotients of hyperbolic 3-manifold groups, Proc. Amer. Math. Soc. 126 (1998), no. 3, 877–880. MR 1459136CrossRefGoogle Scholar
Long, Darren and Reid, Alan W., Surface subgroups and subgroup separability in 3-manifold topology, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2005, 25o Colóquio Brasileiro de Matemática. [25th Brazilian Mathematics Colloquium]. MR 2164951Google Scholar
Lubotzky, Alexander, Some more non-arithmetic rigid groups, Geometry, spectral theory, groups, and dynamics, Contemp. Math., vol. 387, Amer. Math. Soc., Providence, RI, 2005, pp. 237–244. MR 2180210CrossRefGoogle Scholar
Margulis, G. A., Finiteness of quotient groups of discrete subgroups, Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 28–39. MR 545365CrossRefGoogle Scholar
Margulis, G. A., Multiplicative groups of a quaternion algebra over a global field, Dokl. Akad. Nauk SSSR 252 (1980), no. 3, 542–546. MR 577836Google Scholar
Margulis, G. A., Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825CrossRefGoogle Scholar
Martin, Alexandre, On the cubical geometry of Higman’s group, Duke Math. J. 166 (2017), no. 4, 707–738. MR 3619304CrossRefGoogle Scholar
Minasyan, Ashot and Osin, Denis, Acylindrical hyperbolicity of groups acting on trees, Math. Ann. 362 (2015), no. 3-4, 1055–1105. MR 3368093CrossRefGoogle Scholar
Mozes, Shahar, A zero entropy, mixing of all orders tiling system, Symbolic dynamics and its applications (New Haven, CT, 1991), Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 319–325. MR 1185097CrossRefGoogle Scholar
Neumann, B. H., Some remarks on infinite groups., J. Lond. Math. Soc. 12 (1937), 120–127.Google Scholar
Neumann, B. H. and Neumann, Hanna, A contribution to the embedding theory of group amalgams, Proc. London Math. Soc. (3) 3 (1953), 243–256. MR 0057880Google Scholar
Neumann, Peter M., The SQ-universality of some finitely presented groups, J. Austral. Math. Soc. 16 (1973), 1–6, Collection of articles dedicated to the memory of Hanna Neumann, I. MR 0333017CrossRefGoogle Scholar
Niblo, Graham and Reeves, Lawrence, Groups acting on CAT(0) cube complexes, Geom. Topol. 1 (1997), 1–7. MR 1432323CrossRefGoogle Scholar
Ol’shanskiĬ, A. Yu, On residualing homomorphisms and G-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365–409. MR 1250244CrossRefGoogle Scholar
Ol’shanskiĬ, A. Yu, SQ-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132. MR 1357360Google Scholar
Ol’shanskiĬ, A. Yu, On the Bass-Lubotzky question about quotients of hyperbolic groups, J. Algebra 226 (2000), no. 2, 807–817. MR 1752761Google Scholar
Osin, D., Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR 3430352Google Scholar
Platonov, V. P., Arithmetic and structural problems in linear algebraic groups, pp. 471–478, Anad. Math. Ongress, Montreal, Que., 1975. MR 0466334Google Scholar
Platonov, Vladimir and Rapinchuk, Andrei, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994, Translated from the 1991 Russian original by Rachel Rowen. MR 1278263Google Scholar
Radu, Nicolas, New simple lattices in products of trees and their projections, Preprint arXiv:1712.01091.Google Scholar
Radu, Nicolas, A classification theorem for boundary 2-transitive automorphism groups of trees, Invent. Math. 209 (2017), no. 1, 1–60. MR 3660305CrossRefGoogle Scholar
Radu, Nicolas, PhD thesis, Université catholique de Louvain, In preparation, 2017.Google Scholar
Raghunathan, M. S., Discrete subgroups of Lie groups, Math. Student (2007), Special Centenary Volume, 59–70 (2008). MR 2527560Google Scholar
Rapinchuk, Andrei S., The Margulis-Platonov conjecture for SL1,D and 2-generation of finite simple groups, Math. Z. 252 (2006), no. 2, 295–313. MR 2207799CrossRefGoogle Scholar
Rapinchuk, Andrei and Potapchik, Alexander, Normal subgroups of SL1,D and the classification of finite simple groups, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 4, 329–368. MR 1425612Google Scholar
Rapinchuk, Andrei S., Segev, Yoav, and Seitz, Gary M., Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable, J. Amer. Math. Soc. 15 (2002), no. 4, 929–978. MR 1915823CrossRefGoogle Scholar
Rattaggi, Diego, Computations in groups acting on a product of trees: Normal subgroup structures and quaternion lattices, ProQuest LLC, Ann Arbor, MI, 2004, Thesis (Dr.sc.math.)–Eidgenoessische Technische Hochschule Zuerich (Switzerland). MR 2715704Google Scholar
Rattaggi, Diego, Anti-tori in square complex groups, Geom. Dedicata 114 (2005), 189–207. MR 2174099CrossRefGoogle Scholar
Rattaggi, Diego, A finitely presented torsion-free simple group, J. Group Theory 10 (2007), no. 3, 363–371. MR 2320973CrossRefGoogle Scholar
Ronan, M. A., Triangle geometries, J. Combin. Theory Ser. A 37 (1984), no. 3, 294–319. MR 769219CrossRefGoogle Scholar
Rungtanapirom, Nithi, Quaternionic arithmetic lattices of rank 2 and a fake quadric in characteristic 2, Preprint arXiv:1707.09925.Google Scholar
Sageev, Michah, CAT(0) cube complexes and groups, Geometric group theory, IAS/Park City Math. Ser., vol. 21, Amer. Math. Soc., Providence, RI, 2014, pp. 7–54. MR 3329724Google Scholar
Schupp, Paul E., Small cancellation theory over free products with amalgamation, Math. Ann. 193 (1971), 255–264. MR 0291298CrossRefGoogle Scholar
Schupp, Paul E., A survey of small cancellation theory, 569–589. Studies in Logic and the Foundations of Math., Vol. 71. MR 0412289CrossRefGoogle Scholar
Segev, Yoav and Seitz, Gary M., Anisotropic groups of type An and the commuting graph of finite simple groups, Pacific J. Math. 202 (2002), no. 1, 125–225. MR 1883974CrossRefGoogle Scholar
Sela, Zlil, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001), no. 93, 31–105. MR 1863735Google Scholar
Shalom, Yehuda, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), no. 1, 1–54. MR 1767270CrossRefGoogle Scholar
Stix, Jakob and Vdovina, Alina, Simply transitive quaternionic lattices of rank 2 over Fq (t) and a non-classical fake quadric, Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 3, 453–498. MR 3708519CrossRefGoogle Scholar
Tamburini, Chiara and Wilson, John S., A residual property of certain free products, Math. Z. 186 (1984), no. 4, 525–530. MR 744963CrossRefGoogle Scholar
Tits, Jacques, Buildings and group amalgamations, Proceedings of groups—St. Andrews 1985, London Math. Soc. Lecture Note Ser., vol. 121, Cambridge Univ. Press, Cambridge, 1986, pp. 110–127. MR 896503Google Scholar
Tits, Jacques, Résumés des cours au Collège de France 1973–2000, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 12, Société Mathématique de France, Paris, 2013. MR 3235648Google Scholar
Trofimov, V. I. and Weiss, R. M., Graphs with a locally linear group of automorphisms, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 2, 191–206. MR 1341785CrossRefGoogle Scholar
Vignéras, Marie-France, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980. MR 580949CrossRefGoogle Scholar
Wang, Hsien Chung, Topics on totally discontinuous groups, 459–487. Pure and Appl. Math., Vol. 8. MR 0414787Google Scholar
Weiss, Richard, Groups with a (B, N)-pair and locally transitive graphs, Nagoya Math. J. 74 (1979), 1–21. MR 535958CrossRefGoogle Scholar
Wielandt, Helmut, Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. MR 0183775Google Scholar
Wilson, John S., Groups with every proper quotient finite, Proc. Cambridge Philos. Soc. 69 (1971), 373–391. MR 0274575CrossRefGoogle Scholar
Wilson, John S., On just infinite abstract and profinite groups, New horizons in pro-p groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 181–203. MR 1765120Google Scholar
Wilton, Henry, Non-positively curved cube complexes, Course notes, available as AMS Open Math Notes:201704.110697, 2011.Google Scholar
Wilton, Henry, Alternating quotients of free groups, Enseign. Math. (2) 58 (2012), no. 1-2, 49–60. MR 2985009Google Scholar
Wise, Daniel T., Non-positively curved squared complexes: Aperiodic tilings and nonresidually finite groups, ProQuest LLC, Ann Arbor, MI, 1996, Thesis (Ph.D.)– Princeton University. MR 2694733Google Scholar
Wise, Daniel T., Subgroup separability of the figure 8 knot group, Topology 45 (2006), no. 3, 421–463. MR 2218750CrossRefGoogle Scholar
Wise, Daniel T., Complete square complexes, Comment. Math. Helv. 82 (2007), no. 4, 683–724. MR 2341837Google Scholar
Wise, Daniel T., The structure of groups with a quasi-convex hierarchy, Preprint available at https://www.math.u-psud.fr/~haglund/Hierarchy29Feb2012.pdf, 2012.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×