Small cancellations over relatively hyperbolic groups and embedding theorems

Abstract

We generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings. This generalization is then used to prove various embedding theorems for countable groups. For instance, we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes. In particular, this gives the affirmative answer to the well-known question of the existence of a finitely generated group $G$ other than $\mathbb Z/2\mathbb Z$ such that all nontrivial elements of $G$ are conjugate.

  • [ABJ] Go to document G. Arzhantseva, M. R. Bridson, T. Januszkiewicz, I. J. Leary, A. Minasyan, and J. Świcatkowski, "Infinite groups with fixed point properties," Geom. Topol., vol. 13, iss. 3, pp. 1229-1263, 2009.
    @article {ABJ, MRKEY = {2496045},
      AUTHOR = {Arzhantseva, Goulnara and Bridson, Martin R. and Januszkiewicz, Tadeusz and Leary, Ian J. and Minasyan, Ashot and {\'S}wi{\c{a}}tkowski, Jacek},
      TITLE = {Infinite groups with fixed point properties},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry \& Topology},
      VOLUME = {13},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {1229--1263},
      ISSN = {1465-3060},
      MRCLASS = {20F65 (20F67)},
      MRNUMBER = {2496045},
      MRREVIEWER = {Vassilis Metaftsis},
      DOI = {10.2140/gt.2009.13.1229},
      ZBLNUMBER = {05530204},
      }
  • [SQ] Go to document G. Arzhantseva, A. Minasyan, and D. Osin, "The SQ-universality and residual properties of relatively hyperbolic groups," J. Algebra, vol. 315, iss. 1, pp. 165-177, 2007.
    @article {SQ, MRKEY = {2344339},
      AUTHOR = {Arzhantseva, G. and Minasyan, A. and Osin, D.},
      TITLE = {The {SQ}-universality and residual properties of relatively hyperbolic groups},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {315},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {165--177},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20F67},
      MRNUMBER = {2008g:20095},
      MRREVIEWER = {Alexey Muranov},
      DOI = {10.1016/j.jalgebra.2007.04.029},
      ZBLNUMBER = {1132.20022},
      }
  • [Prob] G. Baumslag, A. G. Myasnikov, and V. Shpilrain, "Open problems in combinatorial group theory," in Combinatorial and Geometric Group Theory (New York, 2000/Hoboken, NJ, 2001), Second ed., Providence, RI: Amer. Math. Soc., 2002, pp. 1-38.
    @incollection {Prob, MRKEY = {1921705},
      AUTHOR = {Baumslag, Gilbert and Myasnikov, Alexei G. and Shpilrain, Vladimir},
      TITLE = {Open problems in combinatorial group theory},
      EDITION={Second},
      BOOKTITLE = {Combinatorial and Geometric Group Theory ({N}ew {Y}ork, 2000/{H}oboken, {NJ},
      2001)},
      SERIES = {Contemp. Math.},
      NUMBER = {296},
      PAGES = {1--38},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2002},
      MRCLASS = {20-02 (57Mxx)},
      MRNUMBER = {2003h:20001},
      MRREVIEWER = {Brent Everitt},
      ZBLNUMBER = {1065.20042},
      }
  • [BO] I. Belegradek and D. Osin, "Rips construction and Kazhdan property (T)," Groups Geom. Dyn., vol. 2, iss. 1, pp. 1-12, 2008.
    @article {BO, MRKEY = {2367206},
      AUTHOR = {Belegradek, Igor and Osin, Denis},
      TITLE = {Rips construction and {K}azhdan property ({T})},
      JOURNAL = {Groups Geom. Dyn.},
      FJOURNAL = {Groups, Geometry, and Dynamics},
      VOLUME = {2},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {1--12},
      ISSN = {1661-7207},
      MRCLASS = {20F67 (20F65)},
      MRNUMBER = {2009c:20076},
      MRREVIEWER = {Fran{ç}ois Dahmani},
      ZBLNUMBER = {1152.20039},
      }
  • [Bow] B. H. Bowditch, "Relatively hyperbolic groups," , preprint , 1999.
    @techreport{Bow,
      author={Bowditch, B. H.},
      TITLE={Relatively hyperbolic groups},
      TYPE={preprint},
      YEAR={1999},
      }
  • [BH] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, New York: Springer-Verlag, 1999.
    @book {BH, MRKEY = {1744486},
      AUTHOR = {Bridson, Martin R. and Haefliger, Andr{é}},
      TITLE = {Metric Spaces of Non-Positive Curvature},
      SERIES = {Grundl. Math. Wissen.},
      NUMBER = {319},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1999},
      PAGES = {xxii+643},
      ISBN = {3-540-64324-9},
      MRCLASS = {53C23 (20F65 53C70 57M07)},
      MRNUMBER = {2000k:53038},
      MRREVIEWER = {Athanase Papadopoulos},
      ZBLNUMBER = {0988.53001},
      }
  • [Dah] Go to document F. Dahmani, "Combination of convergence groups," Geom. Topol., vol. 7, pp. 933-963, 2003.
    @article {Dah, MRKEY = {2026551},
      AUTHOR = {Dahmani, Fran{ç}ois},
      TITLE = {Combination of convergence groups},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {7},
      YEAR = {2003},
      PAGES = {933--963},
      ISSN = {1465-3060},
      MRCLASS = {20F67 (20E06)},
      MRNUMBER = {2005g:20063},
      DOI = {10.2140/gt.2003.7.933},
      ZBLNUMBER = {1037.20042},
      }
  • [F] Go to document B. Farb, "Relatively hyperbolic groups," Geom. Funct. Anal., vol. 8, iss. 5, pp. 810-840, 1998.
    @article {F, MRKEY = {1650094},
      AUTHOR = {Farb, B.},
      TITLE = {Relatively hyperbolic groups},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {8},
      YEAR = {1998},
      NUMBER = {5},
      PAGES = {810--840},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {20F32 (20F10 57M07)},
      MRNUMBER = {99j:20043},
      MRREVIEWER = {Michel Coornaert},
      DOI = {10.1007/s000390050075},
      ZBLNUMBER = {0985.20027},
      }
  • [GS] Go to document S. M. Gersten and H. B. Short, "Rational subgroups of biautomatic groups," Ann. of Math., vol. 134, iss. 1, pp. 125-158, 1991.
    @article {GS, MRKEY = {1114609},
      AUTHOR = {Gersten, S. M. and Short, H. B.},
      TITLE = {Rational subgroups of biautomatic groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {134},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {125--158},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {20M05 (20F10 20F32 57M05)},
      MRNUMBER = {92g:20092},
      MRREVIEWER = {Athanase Papadopoulos},
      DOI = {10.2307/2944334},
      ZBLNUMBER = {0744.20035},
      }
  • [GH] Sur les Groupes Hyperboliques d’après Mikhael Gromov, Ghys, E. and de la Harpe, P., Eds., Boston, MA: Birkhäuser, 1990, vol. 83.
    @book {GH, MRKEY = {1086648},
      TITLE = {Sur les Groupes Hyperboliques d'après {M}ikhael {G}romov},
      SERIES = {Progr. Math.},
      VOLUME = {83},
      EDITOR={ Ghys, E. and de la Harpe, P.},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1990},
      PAGES = {xii+285},
      ISBN = {0-8176-3508-4},
      MRCLASS = {53C23 (20F32 57M05 57S30)},
      MRNUMBER = {92f:53050},
      MRREVIEWER = {Viktor Schroeder},
      }
  • [Gro] M. Gromov, "Hyperbolic groups," in Essays in Group Theory, New York: Springer-Verlag, 1987, vol. 8, pp. 75-263.
    @incollection {Gro, MRKEY = {919829},
      AUTHOR = {Gromov, M.},
      TITLE = {Hyperbolic groups},
      BOOKTITLE = {Essays in Group Theory},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      VOLUME = {8},
      PAGES = {75--263},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      MRCLASS = {20F32 (20F06 20F10 22E40 53C20 57R75 58F17)},
      MRNUMBER = {89e:20070},
      MRREVIEWER = {Christopher W. Stark},
      ZBLNUMBER = {0634.20015},
      }
  • [Guba] V. S. Guba, "A finitely generated complete group," Izv. Akad. Nauk SSSR Ser. Mat., vol. 50, iss. 5, pp. 883-924, 1986.
    @article {Guba, MRKEY = {873654},
      AUTHOR = {Guba, V. S.},
      TITLE = {A finitely generated complete group},
      JOURNAL = {Izv. Akad. Nauk {\rm SSSR} Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      NOTE = {In Russian; translated in {\it Math\/}. USSR-{\it Izv\/}. {\bf 29} (1987), 233--277},
      VOLUME = {50},
      YEAR = {1986},
      NUMBER = {5},
      PAGES = {883--924},
      ISSN = {0373-2436},
      MRCLASS = {20F05},
      MRNUMBER = {88e:20034},
      MRREVIEWER = {M. Greendlinger},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0631.20025},
      }
  • [HNN] Go to document G. Higman, B. H. Neumann, and H. Neumann, "Embedding theorems for groups," J. London Math. Soc., vol. 24, pp. 247-254, 1949.
    @article {HNN, MRKEY = {0032641},
      AUTHOR = {Higman, Graham and Neumann, B. H. and Neumann, Hanna},
      TITLE = {Embedding theorems for groups},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {24},
      YEAR = {1949},
      PAGES = {247--254},
      ISSN = {0024-6107},
      MRCLASS = {20.0X},
      MRNUMBER = {11,322d},
      MRREVIEWER = {F. W. Levi},
      DOI = {10.1112/jlms/s1-24.4.247},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0034.30101},
      }
  • [Iva] Go to document S. V. Ivanov, "On some finiteness conditions in semigroup and group theory," Semigroup Forum, vol. 48, iss. 1, pp. 28-36, 1994.
    @article {Iva, MRKEY = {1245903},
      AUTHOR = {Ivanov, Sergei V.},
      TITLE = {On some finiteness conditions in semigroup and group theory},
      JOURNAL = {Semigroup Forum},
      FJOURNAL = {Semigroup Forum},
      VOLUME = {48},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {28--36},
      ISSN = {0037-1912},
      CODEN = {SMGFAN},
      MRCLASS = {20M10},
      MRNUMBER = {94k:20102},
      MRREVIEWER = {D. C. Jackson},
      DOI = {10.1007/BF02573651},
      ZBLNUMBER={0808.20046},
      }
  • [IO0] S. V. Ivanov and A. Y. Ol$’$shanskii, "Some applications of graded diagrams in combinatorial group theory," in Groups—St. Andrews 1989, Vol. 2, Cambridge: Cambridge Univ. Press, 1991, pp. 258-308.
    @incollection {IO0, MRKEY = {1123985},
      AUTHOR = {Ivanov, Sergei V. and Ol$'$shanskii, Alexander Yu.},
      TITLE = {Some applications of graded diagrams in combinatorial group theory},
      BOOKTITLE = {Groups---{S}t. {A}ndrews 1989, {\rm {V}ol. 2}},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      NUMBER = {160},
      PAGES = {258--308},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1991},
      MRCLASS = {20E10 (20E22 20F10 20F32)},
      MRNUMBER = {92j:20022},
      MRREVIEWER = {D. L. Johnson},
      }
  • [Kou] The Kourovka Notebook. Unsolved Problems in Group Theory, fifteenth augmented ed., Mazurov, V. D. and Khukhro, E. I., Eds., Institut Matematiki im. S. L. Soboleva, Novosibirsk: Rossiĭskaya Akademiya Nauk Sibirskoe Otdelenie, 2002.
    @book {Kou, MRKEY = {1956290},
      TITLE = {The {K}ourovka Notebook. Unsolved Problems in Group Theory},
      EDITION = {fifteenth augmented},
      EDITOR={Mazurov, V. D. and Khukhro, E. I. },
      PUBLISHER = {Rossiĭskaya Akademiya Nauk Sibirskoe Otdelenie},
      ADDRESS = {Institut Matematiki im. S. L. Soboleva, Novosibirsk},
      YEAR = {2002},
      PAGES = {164},
      ISBN = {5-94356-060-2},
      MRCLASS = {20-02 (00A07)},
      MRNUMBER = {2004f:20001a},
      }
  • [LS] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, New York: Springer-Verlag, 1977.
    @book {LS, MRKEY = {0577064},
      AUTHOR = {Lyndon, Roger C. and Schupp, Paul E.},
      TITLE = {Combinatorial Group Theory},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {xiv+339},
      ISBN = {3-540-07642-5},
      MRCLASS = {20F05 (55A05)},
      MRNUMBER = {58 \#28182},
      MRREVIEWER = {Ian M. Chiswell},
      ZBLNUMBER = {0368.20023},
      }
  • [MO] K. V. Mikhajlovskii and Y. A. Ol$’$shanskii, "Some constructions relating to hyperbolic groups," in Geometry and Cohomology in Group Theory, Cambridge: Cambridge Univ. Press, 1998, pp. 263-290.
    @incollection {MO, MRKEY = {1709962},
      AUTHOR = {Mikhajlovskii, K. V. and Ol$'$shanskii, A. Yu.},
      TITLE = {Some constructions relating to hyperbolic groups},
      BOOKTITLE = {Geometry and Cohomology in Group Theory},
      VENUE={{D}urham, 1994},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      NUMBER = {252},
      PAGES = {263--290},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1998},
      MRCLASS = {20F67},
      MRNUMBER = {2000h:20083},
      MRREVIEWER = {Panos Papasoglu},
      }
  • [Min] A. Minasyan, "Groups with finitely many conjugacy classes and their automorphisms," Comment. Math. Helv., vol. 84, iss. 2, pp. 259-296, 2009.
    @article {Min, MRKEY = {2495795},
      AUTHOR = {Minasyan, Ashot},
      TITLE = {Groups with finitely many conjugacy classes and their automorphisms},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      VOLUME = {84},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {259--296},
      ISSN = {0010-2571},
      MRCLASS = {20E45 (20F67)},
      MRNUMBER = {2495795},
      MRREVIEWER = {Alexey Muranov},
      ZBLNUMBER = {05545771},
      }
  • [Nor] A. Minasyan and D. Osin, "Normal automorphisms of hyperbolic groups," , preprint , 2008.
    @techreport{Nor,
      author={Minasyan, Ashot and Osin, D.},
      TITLE={Normal automorphisms of hyperbolic groups},
      NOTE={{\it Trans. Amer. Math. Soc.},
      to appear},
      YEAR={2008},
      ARXIV={0809.2408},
      TYPE={preprint},
      }
  • [Ols-book] . A. Ol$’$shanskiui, Geometry of Defining Relations in Groups, Dordrecht: Kluwer Academic Publishers Group, 1991.
    @book {Ols-book, MRKEY = {1191619},
      AUTHOR = {Ol$'$shanski{\u\i},
      A. {Yu}.},
      TITLE = {Geometry of Defining Relations in Groups},
      SERIES = {Mathematics and its Applications (Soviet Series)},
      NUMBER = {70},
      PUBLISHER = {Kluwer Academic Publishers Group},
      ADDRESS = {Dordrecht},
      YEAR = {1991},
      PAGES = {xxvi+505},
      ISBN = {0-7923-1394-1},
      MRCLASS = {20F32 (20E10 57M07)},
      MRNUMBER = {93g:20071},
      ZBLNUMBER={0830.20053},
      }
  • [Ols1] . A. Ol$’$shanskiui, "Periodic quotient groups of hyperbolic groups," Mat. Sb., vol. 182, iss. 4, pp. 543-567, 1991.
    @article {Ols1, MRKEY = {1119008},
      AUTHOR = {Ol$'$shanski{\u\i},
      A. {Yu}.},
      TITLE = {Periodic quotient groups of hyperbolic groups},
      JOURNAL = {Mat. Sb.},
      FJOURNAL = {MatematicheskiĭSbornik},
      VOLUME = {182},
      YEAR = {1991},
      NOTE = {In Russian; translated in {\it Math\/}. USSR {\it Sbornik} {\bf 72} (1992), 519--541},
      NUMBER = {4},
      PAGES = {543--567},
      ISSN = {0368-8666},
      MRCLASS = {20F32 (57M07)},
      MRNUMBER = {92d:20050},
      MRREVIEWER = {Michael Kapovich},
      }
  • [Ols2] Go to document . A. Ol$’$shanskiui, "On residualing homomorphisms and $G$-subgroups of hyperbolic groups," Internat. J. Algebra Comput., vol. 3, iss. 4, pp. 365-409, 1993.
    @article {Ols2, MRKEY = {1250244},
      AUTHOR = {Ol$'$shanski{\u\i},
      A. {Yu}.},
      TITLE = {On residualing homomorphisms and {$G$}-subgroups of hyperbolic groups},
      JOURNAL = {Internat. J. Algebra Comput.},
      FJOURNAL = {International Journal of Algebra and Computation},
      VOLUME = {3},
      YEAR = {1993},
      NUMBER = {4},
      PAGES = {365--409},
      ISSN = {0218-1967},
      MRCLASS = {20F32 (20F05 20F06 57M07)},
      MRNUMBER = {94i:20069},
      MRREVIEWER = {Michel Coornaert},
      DOI = {10.1142/S0218196793000251},
      }
  • [Ols3] Go to document . A. Ol$’$shanskiui, "On the Bass-Lubotzky question about quotients of hyperbolic groups," J. Algebra, vol. 226, iss. 2, pp. 807-817, 2000.
    @article {Ols3, MRKEY = {1752761},
      AUTHOR = {Ol$'$shanski{\u\i},
      A. {Yu}.},
      TITLE = {On the {B}ass-{L}ubotzky question about quotients of hyperbolic groups},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {226},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {807--817},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20F05},
      MRNUMBER = {2001i:20069},
      MRREVIEWER = {Rafael Oswaldo Ruggiero},
      DOI = {10.1006/jabr.1999.8170},
      ZBLNUMBER={0957.20026},
      }
  • [RHG] D. V. Osin, "Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems," Mem. Amer. Math. Soc., vol. 179, iss. 843, p. vi, 2006.
    @article {RHG, MRKEY = {2182268},
      AUTHOR={Osin, D. V.},
      TITLE = {Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems},
      JOURNAL = {Mem. Amer. Math. Soc.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {179},
      YEAR = {2006},
      NUMBER = {843},
      PAGES = {vi+100},
      ISSN = {0065-9266},
      CODEN = {MAMCAU},
      MRCLASS = {20F67},
      MRNUMBER = {2006i:20047},
      MRREVIEWER = {Ilya Kapovich},
      }
  • [ESBG] Go to document D. V. Osin, "Elementary subgroups of relatively hyperbolic groups and bounded generation," Internat. J. Algebra Comput., vol. 16, iss. 1, pp. 99-118, 2006.
    @article {ESBG, MRKEY = {2217644},
      AUTHOR={Osin, D. V.},
      TITLE = {Elementary subgroups of relatively hyperbolic groups and bounded generation},
      JOURNAL = {Internat. J. Algebra Comput.},
      FJOURNAL = {International Journal of Algebra and Computation},
      VOLUME = {16},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {99--118},
      ISSN = {0218-1967},
      MRCLASS = {20F67 (20F65)},
      MRNUMBER = {2007b:20092},
      MRREVIEWER = {Fran{ç}ois Dahmani},
      DOI = {10.1142/S0218196706002901},
      ZBLNUMBER = {1100.20033},
      }
  • [RDF] D. V. Osin, "Relative Dehn functions of amalgamated products and HNN-extensions," in Topological and Asymptotic Aspects of Group Theory, Providence, RI: Amer. Math. Soc., 2006, pp. 209-220.
    @incollection {RDF, MRKEY = {2216718},
      AUTHOR = {Osin, D. V.},
      TITLE = {Relative {D}ehn functions of amalgamated products and {HNN}-extensions},
      BOOKTITLE = {Topological and Asymptotic Aspects of Group Theory},
      SERIES = {Contemp. Math.},
      NUMBER = {394},
      PAGES = {209--220},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2006},
      MRCLASS = {20F65 (20E06 20F05 20F67 57M07)},
      MRNUMBER = {2007g:20042},
      MRREVIEWER = {Thomas Delzant},
      ZBLNUMBER = {1111.20036},
      }
  • [Fac] D. V. Osin, Factorizable groups and finiteness conditions.
    @misc{Fac,
      author={Osin, D. V.},
      TITLE = {Factorizable groups and finiteness conditions},
      NOTE={in preparation},
      }

Authors

Denis Osin

Stevenson Center 1326, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States