Dehn Function - Known Results

Known Results

• A finitely presented group is word-hyperbolic group if and only if its Dehn function is equivalent to n, that is, if and only if every finite presentation of this group satisfies a linear isoperimetric inequality.
• Isoperimetric gap: If a finitely presented group satisfies a subquadratic isoperimetric inequality then it is word-hyperbolic. Thus there are no finitely presented groups with Dehn functions equivalent to nd with d ∈ (1,2).
• Automatic groups and, more generally, combable groups satisfy quadratic isoperimetric inequalities.
• A finitely generated nilpotent group has a Dehn function equivalent to nd where d ≥ 1 and all positive integers d are realized in this way. Moreover, every finitely generated nilpotent group G admits a polynomial isoperimetric inequality of degree c + 1, where c is the nilpotency class of G.
• The set of real numbers d ≥ 1, such that there exists a finitely presented group with Dehn function equivalent to nd, is dense in the interval
• If all asymptotic cones of a finitely presented group are simply connected, then the group satisfies a polynomial isoperimetric inequality.
• If a finitely presented group satisfies a quadratic isoperimetric inequality, then all asymptotic cones of this group are simply connected.
• If (M,g) is a closed Riemannian manifold and G = π₁(M) then the Dehn function of G is equivalent to the filling area function of the manifold.
• If G is a group acting properly discontinuously and cocompactly by isometries on a CAT(0) space, then G satisfies a quadratic isoperimetric inequality. In particular, this applies to the case where G is the fundamental group of a closed Riemannian manifold of non-positive sectional curvature (not necessarily constant).
• The Dehn function of SL(m, Z) is at most exponential for any m ≥ 3. For SL(3,Z) this bound is sharp and it is known in that case that the Dehn function does not admit a subexponential upper bound. The Dehn functions for SL(m,Z), where m > 4 are quadratic. The Dehn function of SL(4,Z), has been conjectured to be quadratic, by Thurston.
• Mapping class groups of surfaces of finite type are automatic and satisfy quadratic isoperimetric inequalities.
• Hatcher and Vogtmann proved that the groups Aut(F_k) and Out(F_k) satisfy exponential isoperimetric inequalities for every k ≥ 3. For k = 3 these bounds are known to be sharp by a result of Bridson and Vogtamann who proved that Aut(F₃) and Out(F₃) do not satisfy subexponential isoperimetric inequalities.
• For every automorphism φ of a finitely generated free group F_k the mapping torus group of φ satisfies a quadratic isoperimetric inequality.
• Most "reasonable" computable functions that are ≥n4, can be realized, up to equivalence, as Dehn functions of finitely presented groups. In particular, if f(n) ≥ n4 is a superadditive function whose binary representation is computable in time by a Turing machine then f(n) is equivalent to the Dehn function of a finitely presented group.
• Although one cannot reasonably bound the Dehn function of a group in terms of the complexity of its word problem, Birget, Olʹshanskii, Rips and Sapir obtained the following result, providing a far-reaching generalization of Higman's embedding theorem: The word problem of a finitely generated group is decidable in nondeterministic polynomial time if and only if this group can be embedded into a finitely presented group with a polynomial isoperimetric function. Moreover, every group with the word problem solvable in time T(n) can be embedded into a group with isoperimetric function equivalent to n2T(n2)4.