Surface subgroups of graph groups
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- by Herman Servatius, Carl Droms and Brigitte Servatius PDF
- Proc. Amer. Math. Soc. 106 (1989), 573-578 Request permission
Abstract:
Given a graph $\Gamma$, define the group ${F_\Gamma }$ to be that generated by the vertices of $\Gamma$, with a defining relation $xy = yx$ for each pair $x,y$ of adjacent vertices of $\Gamma$. In this article, we examine the groups ${F_\Gamma }$, where the graph $\Gamma$ is an $n$-gon, $(n \geq 4)$. We use a covering space argument to prove that in this case, the commutator subgroup ${F’_\Gamma }$ contains the fundamental group of the orientable surface of genus $1 + (n - 4){2^{n - 3}}$. We then use this result to classify all finite graphs $\Gamma$ for which ${F’_\Gamma }$ is a free group.References
- K. I. Appel and P. E. Schupp, Artin groups and infinite Coxeter groups, Invent. Math. 72 (1983), no. 2, 201–220. MR 700768, DOI 10.1007/BF01389320
- P. Cartier and D. Foata, Problèmes combinatoires de commutation et réarrangements, Lecture Notes in Mathematics, No. 85, Springer-Verlag, Berlin-New York, 1969 (French). MR 0239978
- Warren Dicks, An exact sequence for rings of polynomials in partly commuting indeterminates, J. Pure Appl. Algebra 22 (1981), no. 3, 215–228. MR 629331, DOI 10.1016/0022-4049(81)90099-2
- Carl Droms, Subgroups of graph groups, J. Algebra 110 (1987), no. 2, 519–522. MR 910401, DOI 10.1016/0021-8693(87)90063-9
- Ki Hang Kim, L. Makar-Limanov, Joseph Neggers, and Fred W. Roush, Graph algebras, J. Algebra 64 (1980), no. 1, 46–51. MR 575780, DOI 10.1016/0021-8693(80)90131-3
- Ki Hang Kim, L. Makar-Limanov, Joseph Neggers, and Fred W. Roush, Graph algebras, J. Algebra 64 (1980), no. 1, 46–51. MR 575780, DOI 10.1016/0021-8693(80)90131-3
- Stephen J. Pride, On Tits’ conjecture and other questions concerning Artin and generalized Artin groups, Invent. Math. 86 (1986), no. 2, 347–356. MR 856848, DOI 10.1007/BF01389074
- Herman Servatius, Automorphisms of graph groups, J. Algebra 126 (1989), no. 1, 34–60. MR 1023285, DOI 10.1016/0021-8693(89)90319-0
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 106 (1989), 573-578
- MSC: Primary 20F32; Secondary 05C25, 20F34, 57M15
- DOI: https://doi.org/10.1090/S0002-9939-1989-0952322-9
- MathSciNet review: 952322