Mediterranean Journal of Mathematics

, Volume 13, Issue 5, pp 3205–3219 | Cite as

Homomorphisms to \({\mathbb{R}}\) Generated by Quasimorphisms

  • Gastão Bettencourt
  • Sérgio Mendes


Erschler and Karlsson in Annales de l’Institut Fourier 60(6):2095–2113, 2010 construct a homomorphism of a finitely generated group G to \({\mathbb{R}}\) using a random walk approach. Central to their construction were the word length \({\ell}\) and a well behaved measure \({\mu}\) on G. We consider a modified version of this construction using instead of \({\ell}\) a quasimorphism f of G. Moreover, if a group H acts on G via group automorphisms we show that this technique can be adapted to construct a homomorphism of the semidirect product \({G \rtimes_{\phi} H}\) to \({\mathbb{R}}\), in analogy with the word length case (Bettencourt and Mendes, Appl. Math. Inf. Sci. 9(6):1–7, 2015).


Random walks on groups Homomorphisms Quasimorphisms Semidirect product 

Mathematics Subject Classification

Primary 60B15 Secondary 20F65 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversidade da Beira InteriorCovilhãPortugal
  2. 2.Department of MathematicsISCTE-Lisbon University InstituteLisbonPortugal

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