A finitely presented torsion-free simple group
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Diego Rattaggi
Abstract
We construct a finitely presented torsion-free simple group Σ0, acting cocompactly on a product of two regular trees. An infinite family of such groups was introduced by Burger and Mozes [M. Burger and S. Mozes. Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 747–752.], [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92 (2001), 151–194.]. We refine their methods and construct Σ0 as an index 4 subgroup of a group 
presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92 (2001), 151–194., Theorem 6.4] needs more than 18000 relations, and the smallest simple group constructed in [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92 (2001), 151–194., §6.5] needs even more than 360000 relations in any finite presentation.
© Walter de Gruyter
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- Soluble normally constrained pro-p-groups
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Articles in the same Issue
- Transitivity properties for group actions on buildings
- Fields, values and character extensions in finite groups
- Primitive sharp permutation groups with large solvable subgroups
- On the size of the nilpotent residual in finite groups
- On the commuting complex of finite metanilpotent groups
- Soluble normally constrained pro-p-groups
- Comparing quasi-finitely axiomatizable and prime groups
- A finitely presented torsion-free simple group
- Andrews–Curtis groups and the Andrews–Curtis conjecture
- Residual finiteness of outer automorphism groups of certain tree products
- Commutator subgroups of Hantzsche–Wendt groups
