## The geometry of one-relator groups satisfying a polynomial isoperimetric inequality

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- by Giles Gardam and Daniel J. Woodhouse PDF
- Proc. Amer. Math. Soc.
**147**(2019), 125-129 Request permission

## Abstract:

For every pair of positive integers $p > q$ we construct a one-relator group $R_{p,q}$ whose Dehn function is $\simeq n^{2 \alpha }$ where $\alpha = \log _2(2p / q)$. The group $R_{p, q}$ has no subgroup isomorphic to a Baumslag–Solitar group $BS(m, n)$ with $m \neq \pm n$, but it is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a long-standing question on the automaticity of one-relator groups and gives counterexamples to a conjecture of Wise.## References

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## Additional Information

**Giles Gardam**- Affiliation: Department of Mathematics, Technion, Haifa, Israel
- MR Author ID: 1248225
- Email: gilesgar@technion.ac.il
**Daniel J. Woodhouse**- Affiliation: Department of Mathematics, Technion, Haifa, Israel
- MR Author ID: 1175525
- Email: woodhouse.da@technion.ac.il
- Received by editor(s): December 8, 2017
- Received by editor(s) in revised form: May 3, 2018
- Published electronically: October 18, 2018
- Additional Notes: The first author was supported by the Israel Science Foundation (grant 662/15).

The second author was supported by the Israel Science Foundation (grant 1026/15). - Communicated by: David Futer
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 125-129 - MSC (2010): Primary 20F65; Secondary 20F67, 20E06, 20F05
- DOI: https://doi.org/10.1090/proc/14238
- MathSciNet review: 3876736