Decomposing finitely generated groups into free products with amalgamation

V. V. Benyash-Krivets

Sbornik. Mathematics, Tome 192 (2001) no. 2, pp. 163-186

Résumé

The problem of the existence of a decomposition of a finitely generated group $\Gamma$ into a non-trivial free product with amalgamation is studied. It is proved that if $\dim X^s(\Gamma )\geqslant 2$, where $X^s(\Gamma )$ is the character variety of irreducible representations of $\Gamma$ into $\operatorname {SL}_2(\mathbb C)$, then $\Gamma$ is a non-trivial free product with amalgamation. Next, the case when $\Gamma =\langle a,b\mid a^n=b^k=R^m(a,b)\rangle $ is a generalized triangle group is considered. It is proved that if one of the generators of $\Gamma$ has infinite order, then $\Gamma$ is a non-trivial free product with amalgamation. In the general case sufficient conditions ensuring that $\Gamma$ is a non-trivial free product with amalgamation are found.

Bibliographie

[1] Lindon R., Shupp P., Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[2] Wall C. T. C. (ed.), Homological group theory, Proc. of a conference in Durham, London Math. Soc. Lecture Note Ser., 36, 1977

[3] Baumslag G., Shalen P. B., “Amalgamated products and finitely presented groups”, Comment. Math. Helv., 65 (1990), 243–254 | DOI | MR | Zbl

[4] Fine B., Levin F., Rosenberger G., “Free subgroups and decompositions of one-relator products of cyclics. 2: Normal torsion free subgroups and FPA decompositions”, J. Indian Math. Soc., 49 (1985), 237–247 | MR

[5] Zieschang H., “On decompositions of discontinuous groups of the plane”, Math. Z., 151 (1976), 165–188 | DOI | MR | Zbl

[6] Rosenberger G., “Bemerkungen zu einer Arbeit von H. Zieschang”, Arch. Math. (Basel), 29 (1977), 623–627 | MR | Zbl

[7] Long D. D., Maclachlan C., Reid A. W., “Splitting groups of signature $(1,n)$”, J. Algebra, 185 (1996), 329–341 | DOI | MR | Zbl

[8] Dunwoody M. J., Sageev M., “Splittings of certain Fuchsian groups”, Proc. Amer. Math. Soc., 125:7 (1997), 1953–1954 | DOI | MR | Zbl

[9] Benyash-Krivets V. V., “O razlozhenii svobodnogo proizvedeniya tsiklicheskikh grupp s odnim sootnosheniem v amalgamirovannoe svobodnoe proizvedenie”, Matem. sb., 189:8 (1998), 13–26 | MR | Zbl

[10] Benyash-Krivets V. V., “O razlozhimosti nekotorykh $F$-grupp v amalgamirovannoe svobodnoe proizvedenie”, Dokl. ANB, 41:6 (1997), 1–4 | MR

[11] Lubotzky A., Magid A., “Varieties of representations of finitely generated groups”, Memoirs Amer. Math. Soc., 58 (1985), 1–116 | MR

[12] Culler M., Shalen P., “Varieties of group representations and splittings of 3 manifolds”, Ann. of Math., 117 (1983), 109–147 | DOI | MR

[13] Rapinchuk A. S., Benyash-Krivetz V. V., Chernousov V. I., “Representation varieties of the fundamental groups of compact orientable surfaces”, Israel J. Math., 93 (1996), 29–71 | DOI | MR | Zbl

[14] Benyash-Krivets V. V., Chernousov V. I., “Mnogoobraziya predstavlenii fundamentalnykh grupp kompaktnykh neorientiruemykh poverkhnostei”, Matem. sb., 188:7 (1997), 47–92 | MR | Zbl

[15] Mamford D., Algebraicheskaya geometriya, Mir, M., 1979 | MR

[16] Horowitz R., “Characters of free groups represented in the two dimensional linear group”, Comm. Pure. Appl. Math., 25 (1972), 635–649 | DOI | MR

[17] Helling H., “Diskrete Untergruppen von $\operatorname{SL}_2(\mathbb R)$”, Invent. Math., 17 (1972), 217–229 | DOI | MR | Zbl

[18] Magnus W., “The uses of 2 by 2 matrices in combinatorial group theory”, Results Math., 4:2 (1981), 171–192 | MR | Zbl

[19] Serre J.-P., Arbres, amalgames, $\operatorname{SL}_2$, Astérisque, 46, 1977 | MR | Zbl

[20] Burbaki N., Gruppy i algebry Li, Mir, M., 1976 | MR

[21] Majeed A., Mason A. W., “Solvable-by-finite subgroups of $\operatorname{GL}(2,F)$”, Glasgow Math. J., 19 (1978), 45–48 | DOI | MR | Zbl

[22] Serr Zh.-P., Algebry Li i gruppy Li, Mir, M., 1969 | MR | Zbl

[23] Vinberg E. B., “Koltsa opredeleniya plotnykh podgrupp poluprostykh lineinykh grupp”, Izv. AN SSSR. Ser. matem., 35:1 (1971), 45–55 | MR | Zbl

[24] Shafarevich I. R., Osnovy algebraicheskoi geometrii, Nauka, M., 1972 | MR | Zbl

[25] Traina C., “Trace polynomial for two generated subgroups of $\operatorname{SL}_2(\mathbb C)$”, Proc. Amer. Math. Soc., 79 (1980), 369–372 | DOI | MR

[26] Bass H., “Finitely generated subgroups of $\operatorname{GL}_2(\mathbb C)$”, The Smith Conjecture, Wiley, New York, 1984, 127–136 | MR