The scale function and lattices

Willis, G.

doi:10.1090/proc/13449

The scale function and lattices
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by G. A. Willis

Proc. Amer. Math. Soc. 145 (2017), 3185-3190

DOI: https://doi.org/10.1090/proc/13449

Published electronically: January 23, 2017

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Abstract:

It is shown that, given a lattice $H$ in a totally disconnected, locally compact group $G$, the contraction subgroups in $G$ and the values of the scale function on $G$ are determined by their restrictions to $H$. Group theoretic properties intrinsic to the lattice, such as being periodic or infinitely divisible, are then seen to imply corresponding properties of $G$.

References

Uri Bader, Pierre-Emmanuel Caprace, Tsachik Gelander, and Shahar Mozes, Simple groups without lattices, Bull. Lond. Math. Soc. 44 (2012), no. 1, 55–67. MR 2881324, DOI 10.1112/blms/bdr061
Hyman Bass and Alexander Lubotzky, Tree lattices, Progress in Mathematics, vol. 176, Birkhäuser Boston, Inc., Boston, MA, 2001. With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. Tits. MR 1794898, DOI 10.1007/978-1-4612-2098-5
Udo Baumgartner, Scales for co-compact embeddings of virtually free groups, Geom. Dedicata 130 (2007), 163–175. MR 2365784, DOI 10.1007/s10711-007-9212-2
Udo Baumgartner, Jacqui Ramagge, and George A. Willis, Scale-multiplicative semigroups and geometry: automorphism groups of trees, Groups Geom. Dyn. 10 (2016), no. 3, 1051–1075. MR 3551189, DOI 10.4171/GGD/376
Udo Baumgartner and George A. Willis, Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Israel J. Math. 142 (2004), 221–248. MR 2085717, DOI 10.1007/BF02771534
Meenaxi Bhattacharjee and Dugald MacPherson, Strange permutation representations of free groups, J. Aust. Math. Soc. 74 (2003), no. 2, 267–285. MR 1957883, DOI 10.1017/S1446788700003293
Adrien Le Boudec, Groups acting on trees with almost prescribed local action, Comment. Math. Helv. 91 (2016), no. 2, 253–293. MR 3493371, DOI 10.4171/CMH/385
Marc Burger and Shahar Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000), 151–194 (2001). MR 1839489, DOI 10.1007/BF02698916
Pierre-Emmanuel Caprace and Nicolas Monod, Decomposing locally compact groups into simple pieces, Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 1, 97–128. MR 2739075, DOI 10.1017/S0305004110000368
Pierre-Emmanuel Caprace, Colin D. Reid, and George A. Willis, Limits of contraction groups and the Tits core, J. Lie Theory 24 (2014), no. 4, 957–967. MR 3328731
Pierre-Emmanuel Caprace, Colin D. Reid, and George A. Willis, Locally normal subgroups of simple locally compact groups, C. R. Math. Acad. Sci. Paris 351 (2013), no. 17-18, 657–661 (English, with English and French summaries). MR 3124321, DOI 10.1016/j.crma.2013.09.010
D. van Dantzig, Zur topologischen Algebra, Math. Ann. 107 (1933), no. 1, 587–626 (German). MR 1512818, DOI 10.1007/BF01448911
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
Wojciech Jaworski, On contraction groups of automorphisms of totally disconnected locally compact groups, Israel J. Math. 172 (2009), 1–8. MR 2534235, DOI 10.1007/s11856-009-0059-0
A. Kepert and G. Willis, Scale functions and tree ends, J. Aust. Math. Soc. 70 (2001), no. 2, 273–292. MR 1815284, DOI 10.1017/S1446788700002640
Alexander Lubotzky, Lattices in rank one Lie groups over local fields, Geom. Funct. Anal. 1 (1991), no. 4, 406–431. MR 1132296, DOI 10.1007/BF01895641
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
Yehuda Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), no. 1, 1–54. MR 1767270, DOI 10.1007/s002220000064
G. Willis, The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), no. 2, 341–363. MR 1299067, DOI 10.1007/BF01450491
George A. Willis, Further properties of the scale function on a totally disconnected group, J. Algebra 237 (2001), no. 1, 142–164. MR 1813900, DOI 10.1006/jabr.2000.8584