Low-dimensional bounded cohomology and extensions of groups
Bounded cohomology of groups was first studied by Gromov in 1982 in his seminal paper M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982), no. 56, 5–99. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic “non-positively curved” groups. On the other hand, there is a well-known interpretation of ordinary group cohomology in dimension $2$ and $3$ in terms of group extensions. The aim of this paper is to make this interpretation available for bounded group cohomology. This will involve quasihomomorphisms as defined and studied by K. Fujiwara and M. Kapovich, On quasihomomorphisms with noncommutative targets, Geom. Funct. Anal. 26 (2016), no. 2, 478–519.
Bucher, M., Frigerio, R., and Hartnick, T., A note on semi-conjugacy for circle actions, Enseign. Math. 62 (2016), no. 3-4, 317–360. https://doi.org/10.4171/LEM/62-3/4-1
Calegari, D., scl, MSJ Memoirs, vol. 20, Mathematical Society of Japan, Tokyo, 2009. https://doi.org/10.1142/e018
Frigerio, R., Bounded cohomology of discrete groups, Mathematical Surveys and Monographs, vol. 227, American Mathematical Society, Providence, RI, 2017.
Frigerio, R., Pozzetti, M. B., and Sisto, A., Extending higher-dimensional quasi-cocycles, J. Topol. 8 (2015), no. 4, 1123–1155. https://doi.org/10.1112/jtopol/jtv017
Fujiwara, K. and Kapovich, M., On quasihomomorphisms with noncommutative targets, Geom. Funct. Anal. 26 (2016), no. 2, 478–519. https://doi.org/10.1007/s00039-016-0364-9
Ghys, É., Groupes d'homéomorphismes du cercle et cohomologie bornée, in “The Lefschetz centennial conference, Part III (Mexico City, 1984)”, Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987, pp. 81–106.
Ghys, É., Groups acting on the circle, Monograf\'ıas del Instituto de Matemática y Ciencias Afines, vol. 6, Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999, a paper from the 12th Escuela Latinoamericana de Matemáticas (XII-ELAM) held in Lima, June 28–July 3, 1999.
Gromov, M., Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982), no. 56, 5–99.
Hartnick, T. and Schweitzer, P., On quasioutomorphism groups of free groups and their transitivity properties, J. Algebra 450 (2016), 242–281. https://doi.org/10.1016/j.jalgebra.2015.10.010
Hull, M. and Osin, D., Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebr. Geom. Topol. 13 (2013), no. 5, 2635–2665. https://doi.org/10.2140/agt.2013.13.2635
MacLane, S., Cohomology theory in abstract groups. III. Operator homomorphisms of kernels, Ann. of Math. (2) 50 (1949), 736–761. https://doi.org/10.2307/1969561
MacLane, S., Homology, first ed., Die Grundlehren der mathematischen Wissenschaften, vol. 114, Springer-Verlag, Berlin-New York, 1967.
Mineyev, I., Bounded cohomology characterizes hyperbolic groups, Q. J. Math. 53 (2002), no. 1, 59–73. https://doi.org/10.1093/qjmath/53.1.59
Monod, N., An invitation to bounded cohomology, in “International Congress of Mathematicians. Vol. II”, Eur. Math. Soc., Zürich, 2006, pp. 1183–1211.
Soma, T., Bounded cohomology and topologically tame Kleinian groups, Duke Math. J. 88 (1997), no. 2, 357–370. https://doi.org/10.1215/S0012-7094-97-08814-1