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Coarse equivalence and topological couplings of locally compact groups

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Abstract

M. Gromov has shown that any two finitely generated groups \(\Gamma \) and \(\Lambda \) are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions \(\Gamma \curvearrowright X \curvearrowleft \Lambda \) on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact second-countable groups.

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Notes

  1. The G-action written on the left will be a left-action, while the H-action written on the right will be a right-action. However, both groups G and H will be equipped with their left-invariant coarse structure, which is that induced by a compatible proper left-invariant metric.

References

  1. Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, pp. 1–295. Cambridge University Press, Cambridge (1993)

  2. Kakutani, S., Kodaira, K.: Über das Haarsche Mass in der lokal bikompakten Gruppe. Proc. Imp. Acad. Tokyo 20, 444–450 (1944)

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  3. Koivisto, J., Kyed, D., Raum, S.: Measure equivalence and coarse equivalence for locally compact amenable groups. arXiv:1703.08121

  4. Rosendal, C.: Coarse geometry of topological groups, manuscript (2017)

  5. Struble, R.A.: Metrics in locally compact groups. Compos. Math. 28, 217–222 (1974)

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, 234 Herzl Street, Rehovot, 7610001, Israel

    Uri Bader

  2. Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL, 60607-7045, USA

    Christian Rosendal

  3. Department of Mathematics, University of Maryland, 4176 Campus Drive - William E. Kirwan Hall, College Park, MD, 20742-4015, USA

    Christian Rosendal

Authors
  1. Uri Bader
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  2. Christian Rosendal
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Corresponding author

Correspondence to Christian Rosendal.

Additional information

U. B. was partially supported by the ISF-Moked Grant 2095/15 and the ERC Grant 306706, C. R. was partially supported by the NSF (DMS 1464974). The authors are grateful for the helpful suggestions by the referee.

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Bader, U., Rosendal, C. Coarse equivalence and topological couplings of locally compact groups. Geom Dedicata 196, 1–9 (2018). https://doi.org/10.1007/s10711-017-0300-7

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  • DOI: https://doi.org/10.1007/s10711-017-0300-7

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