Invariant Schreier decorations of unimodular random networks
Annales Henri Lebesgue, Volume 4 (2021), pp. 1705-1726.

KeywordsSchreier graph, graph limits, unimodular random graph, Invariant Random Subgroup

### Abstract

We prove that every $2d$-regular unimodular random network carries an invariant random Schreier decoration. Equivalently, it is the Schreier coset graph of an invariant random subgroup of the free group ${F}_{d}$. As a corollary we get that every $2d$-regular graphing is the local isomorphic image of a graphing coming from a p.m.p. action of ${F}_{d}$.

The key ingredients of the analogous statement for finite graphs do not generalize verbatim to the measurable setting. We find a more subtle way of adapting these ingredients and prove measurable coloring theorems for graphings along the way.

### References

[AGV14] Abért, Miklós; Glasner, Yair; Virág, Bálint Kesten’s theorem for invariant random subgroups, Duke Math. J., Volume 163 (2014) no. 3, pp. 465-488 | MR 3165420 | Zbl 1344.20061

[AL07] Aldous, David J.; Lyons, Russell Processes on unimodular random networks, Electron. J. Probab., Volume 12 (2007), pp. 1454-1508 | MR 2354165 | Zbl 1131.60003

[Bow12] Bowen, Lewis Invariant random subgroups of the free group (2012) (https://arxiv.org/abs/1204.5939v1)

[BS01] Benjamini, Itai; Schramm, Oded Recurrence of Distributional Limits of Finite Planar Graphs, Electron. J. Probab., Volume 6 (2001), 23, p. 13 pp. | Article | MR 1873300 | Zbl 1010.82021

[BT17] Biringer, Ian; Tamuz, Omer Unimodularity of invariant random subgroups, Trans. Am. Math. Soc., Volume 369 (2017) no. 6, pp. 4043-4061 | Article | MR 3624401 | Zbl 1365.28014

[Can13] Cannizzo, Jan On invariant Schreier structures (2013) (https://arxiv.org/abs/1309.5163v1)

[CK13] Conley, Clinton T.; Kechris, Alexander S. Measurable chromatic and independence numbers for ergodic graphs and group actions, Groups Geom. Dyn., Volume 7 (2013) no. 1, pp. 127-180 | Article | MR 3019078 | Zbl 1315.03082

[CL17] Csóka, Endre; Lippner, Gabor Invariant random perfect matchings in Cayley graphs, Groups Geom. Dyn., Volume 11 (2017) no. 1, pp. 211-244 | Article | MR 3641840 | Zbl 1369.05102

[CLP16] Csóka, Endre; Lippner, Gabor; Pikhurko, Oleg Kőnig’s line coloring and Vizing’s theorems for graphings, Forum Math. Sigma, Volume 4 (2016), e27 | Zbl 1358.05099

[Ele07] Elek, Gábor On limits of finite graphs, Combinatorica, Volume 27 (2007) no. 4, pp. 503-507 | Article | MR 2359831 | Zbl 1164.05061

[HLS14] Hatami, Hamed; Lovász, László; Szegedy, Balázs Limits of locally–globally convergent graph sequences, Geom. Funct. Anal., Volume 24 (2014) no. 1, pp. 269-296 | Article | MR 3177383 | Zbl 1294.05109

[KM15] Kechris, Alexander S.; Marks, Andrew S. Descriptive graph combinatorics, 2015 (preprint available at http://www.math.caltech.edu/~kechris/papers/combinatorics20book.pdf)

[KST99] Kechris, Alexander S.; Solecki, Slawomir; Todorcevic, Stevo Borel chromatic numbers, Adv. Math., Volume 141 (1999) no. 1, pp. 1-44 | Article | MR 1667145 | Zbl 0918.05052

[Lac88] Laczkovich, Miklós Closed sets without measurable matching, Proc. Am. Math. Soc., Volume 103 (1988) no. 3, pp. 894-896 | Article | MR 947676 | Zbl 0668.28002

[Lov12] Lovász, László Large networks and graph limits, Colloquium Publications, 60, American Mathematical Society, 2012 | Zbl 1292.05001