Geometriae Dedicata

, Volume 175, Issue 1, pp 267–280 | Cite as

Relative second bounded cohomology of free groups

  • Cristina Pagliantini
  • Pascal Rolli
Original Paper


This paper is devoted to the computation of the space \(\mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\), where \(\Gamma \) is a free group of finite rank \(n\ge 2\) and \(H\) is a subgroup of finite rank. More precisely we prove that \(H\) has infinite index in \(\Gamma \) if and only if \(\mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\) is not trivial, and furthermore, if and only if there is an isometric embedding \(\oplus _\infty ^n\mathcal {D}({\mathbb {Z}})\hookrightarrow \mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\), where \(\mathcal {D}({\mathbb {Z}})\) is the space of bounded alternating functions on \({\mathbb {Z}}\) equipped with the defect norm.


Relative bounded cohomology Quasimorphisms Free groups  Split quasimorphisms Schreier graphs 

Mathematics Subject Classification

20J06 20E05 55N10 57M15 



We would like to thank Alessandro Sisto for pointing out how the results in [9] imply a weak version of our main theorem. Both authors were supported by Swiss National Science Foundation Project 144373, moreover the second author received support by Project 127016.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department MathematikETH ZürichZurichSwitzerland

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