Homological and homotopical Dehn functions are different
The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, whereas the homotopical Dehn function measures fillings of curves by disks. Because the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions; however, before this work, there were no known examples of finitely presented groups for which the two functions differ. This paper gives such examples, constructed by amalgamating a free-by-cyclic group with several Bestvina-Brady groups.
Publication Source (Journal or Book title)
Proceedings of the National Academy of Sciences of the United States of America
Abrams, A., Brady, N., Dani, P., & Young, R. (2013). Homological and homotopical Dehn functions are different. Proceedings of the National Academy of Sciences of the United States of America, 110 (48), 19206-19212. https://doi.org/10.1073/pnas.1207377110