Thickness of Out(A₁*...*A_n)

2018/11/01 by Saikat Das, Das, Saikat
Mathematics · #20E06 #20E08 #20F28 #20F65 #20F69 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E06 #msc:20E08 #msc:20F28 #msc:20F65 #msc:20F69

paper · pdf · doi:10.48550/arxiv.1811.00435

arxiv created 2018/11/03 · arxiv updated 2018/11/06

Abstract

In this paper we have examined hyperbolicity and relative hyperbolicity of Γ_n := Out(G_n) , where G_n = A₁*...*A_n, is a finite free product and each A_i is a finite group. We have used the Out(G_n) action on the Guirardel-Levitt deformation space, to find a virtual generating set and prove quasi isometric embedding of a large class of subgroups. We have used ideas from works of Mosher-Handel and Alibegović to prove non-distortion. We have used these subgroups to prove that Γ_n is thick for higher complexities. Thickness was developed by Behrstock-Druţu-Mosher and thickness implies that the groups are non-relatively hyperbolic.

Related

• Relative hyperbolicity, thickness, and the hierarchically hyperbolic boundary
• Property (QT) of relatively hierarchically hyperbolic groups
• Relatively hyperbolic groups with fixed peripherals
• Divergence, thick groups, and short conjugators
• Action rigidity for free products of hyperbolic manifold groups