2013/12/01 by Mathieu Carette, Carette, Mathieu
Mathematics · #20E08 #20E42 #20F65 #22D05 #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #math.GR #math.MG #msc:20E08 #msc:20E42 #msc:20F65 #msc:22D05
paper · pdf · doi:10.48550/arxiv.1312.0278
22 pages. Final version, incorporating referee's comments. To appear in Annales de L'Institut Fourier
arxiv created 2014/12/17 · arxiv updated 2014/12/18
Commability is the finest equivalence relation between locally compact groups such that G and H are equivalent whenever there is a continuous proper homomorphism G → H with cocompact image. Answering a question of Cornulier, we show that all non-elementary locally compact groups acting geometrically on locally finite simplicial trees are commable, thereby strengthening previous forms of quasi-isometric rigidity for trees. We further show that 6 homomorphisms always suffice, and provide the first example of a pair of locally compact groups which are commable but without commation consisting of less than 6 homomorphisms. Our strong quasi-isometric rigidity also applies to products of symmetric spaces and Euclidean buildings, possibly with some factors being trees.