Rad HAZU, Matematičke znanosti, Vol. 29 (2025), 283-297.

A HUREWICZ-TYPE FORMULA FOR ASYMPTOTIC-DIMENSION-LOWERING SYMMETRIC QUASIMORPHISMS OF COUNTABLE APPROXIMATE GROUPS

Vera Tonić

Abstract.   A well-known Hurewicz-type formula for asymptotic-dimension-lowering group homomorphisms, due to A. Dranishnikov and J. Smith, states that if f : G → H is a group homomorphism, then asdim G ≤ asdim H + asdim (ker f). In this paper we establish a similar formula for certain quasimorphisms of countable approximate groups: if (Ξ, Ξ^∞) and (Λ, Λ^∞) are countable approximate groups and if f : (Ξ, Ξ^∞) → (Λ, Λ^∞) is a symmetric unital quasimorphism, we show that asdim Ξ ≤ asdim Λ + asdim (f ^-1(D(f))), where D(f) is the defect set of f.

2020 Mathematics Subject Classification.   Primary: 51F30, 20F69; Secondary: 20N99.

Key words and phrases.   Asymptotic dimension, approximate group, quasimorphism.

DOI: https://doi.org/10.21857/yvjrdcdjly

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