Coarse structures on groups defined by conjugations

For a group G, we denote by G↔ the coarse space on G endowed with structure base {{(x,y)∈G×G:y∈xF}:F∈[G]<ω}, xF={z−1xz:z∈F}. Our goal is to explore interplays between algebraic properties of and asymptotic G↔. In particular, show that asdim G↔=0 if only G/ZG locally finite, ZG center G. an infinite subgroups discrete Dedekind group.