Isoperimetric inequalities for the fundamental groups of torus bundles over the circle

Non-positive curvature and complexity for finitely presented groups

“…Certain combable groups have cubic Dehn functions (3.2), as does the 3-dimensional Heisenberg group. In about 1992, sequences of groups (Γ d ) d∈N such that the Dehn function of Γ d is polynomial of degree d were discovered by Gromov [64], Baumslag-Miller-Short [7], and Bridson-Pittet [42]. The literature now contains such sequences with all manner of additional properties.…”

Fractional isoperimetric inequalities and subgroup distortion

“…Assertion (1) is proved in [11] (see also 5A 2 of [14] and [9]). Upper bounds are obtained on Dehn functions in [11] by using the combings of G c constructed in [3].…”

“…Thus we may remove all monochromatic regions of type R 1 from ∆, and applying our inductive hypothesis to the resulting collection of van Kampen diagrams (considered as diagrams over the subpresentation described in the first paragraph of the proof) we deduce that all monochromatic regions of ∆ are homeomorphic to discs. The Dehn functions for many groups of the form Z m Z were calculated in [11], and this classification was later extended to all abelian-by-free groups (see [8], [4]). We shall be concerned with a particular class of examples from [11], namely the groups G c = Z c φc Z, where φ c ∈ GL(c, Z) is the unipotent matrix with ones on the diagonal and super-diagonal and zeros elsewhere.…”

“…The Dehn functions for many groups of the form Z m Z were calculated in [11], and this classification was later extended to all abelian-by-free groups (see [8], [4]). We shall be concerned with a particular class of examples from [11], namely the groups G c = Z c φc Z, where φ c ∈ GL(c, Z) is the unipotent matrix with ones on the diagonal and super-diagonal and zeros elsewhere. G c has presentation:…”

“…About seven years ago, as the result of efforts by a number of authors [9], [11], [14], it was established that for every positive integer d one can construct finitely presented groups whose Dehn function is polynomial of degree d. The question of whether or not there exist groups whose Dehn functions are of fractional degree has attracted a good deal of interest (e.g., [18], [25]). According to a theorem of M. Gromov, one does not get fractional exponents less than 2, because if a group satisfies a sub-quadratic isoperimetric inequality, then it actually satisfies a linear isoperimetric inequality (see [13], [10], [19], [22]).…”