Groups, Geometry, and Dynamics


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Volume 11, Issue 1, 2017, pp. 211–243
DOI: 10.4171/GGD/395

Published online: 2017-04-20

Invariant random perfect matchings in Cayley graphs

Endre Csóka[1] and Gabor Lippner[2]

(1) University of Warwick, Coventry, UK
(2) Northeastern University, Boston, USA

We prove that every non-amenable Cayley graph admits a factor of IID perfect matching. We also show that any connected $d$-regular vertex transitive graph admits a perfect matching. The two results together imply that every Cayley graph admits an invariant random perfect matching.

A key step in the proof is a result on graphings that also applies to finite graphs. The finite version says that for any partial matching of a finite regular graph that is a good expander, one can always find an augmenting path whose length is poly-logarithmic in one over the ratio of unmatched vertices.

Keywords: Perfect matching, factor of IID

Csóka Endre, Lippner Gabor: Invariant random perfect matchings in Cayley graphs. Groups Geom. Dyn. 11 (2017), 211-243. doi: 10.4171/GGD/395