Growth rate of endomorphisms of Houghton's groups

2015/11/30 by Jong Bum Lee, Lee, Jong Bum, Sang Rae Lee +1
Mathematics · #20E36 #20F28 #20K30 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E36 #msc:20F28 #msc:20K30

paper · pdf · doi:10.48550/arxiv.1512.00079

36 pages, 4 figures

arxiv created 2015/11/30 · arxiv updated 2015/12/02

Abstract

A Houghton's group H_n consists of translations at infinity of a n rays of discrete points on the plane. In this paper we study the growth rate of endomorphisms of Houghton's groups. We show that if the kernel of an endomorphism ϕ is not trivial then the growth rate GR(ϕ) equals either 1 or the spectral radius of the induced map on the abelianization. It turns out that every monomorphism ϕ of H_n determines a unique natural number ℓ such that ϕ(H_n) is generated by translations with the same translation length ℓ. We use this to show that GR(ϕ) of a monomorphism ϕ of H_n is precisely ℓ for all 2≤ n.

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