Hyperbolic groups are exactly the finitely presented groups with linear Dehn function, so a non-hyperbolic subgroup of one must fill loops more expensively - but until now no example came with an explicit exponent. This Idea covers the first such bound: Brady's finitely presented non-hyperbolic subgroup H of a hyperbolic group G, obtained by fibring a non-positively curved cube complex over the circle, satisfies n^2 <= delta_H(n) <= n^96. The argument replaces Gersten and Short's algebraic bookkeeping, which needs presentations nobody can write down in practice, with a geometric one: fill a loop in the universal cover at area n log n using thin triangles, then push the filling down one level set of the Morse function at a time, filling the resulting short loops inside the simply connected ascending and descending links. Each push multiplies area by at most C3 = 3T + 1, giving area n^(1 + delta log2 C3) log2 n; for Brady's complex the links are suspensions of a 20-cycle or a 4-cycle, so T <= 20, C3 = 61, and delta = 16, producing the exponent 96. The delta comes from a second result of independent interest: the 1-skeleton of the universal cover is a median graph, and a median graph's optimal 4-point hyperbolicity constant equals the side of its largest isometrically embedded square grid, which here is 4. A student should be able to say why the exponent is an upper bound rather than the truth, where each of the three constants enters, and which steps the same method reaches in the analogous examples of Lodha and Kropholler.
A geometric-group-theory research paper (21 pages, 19 figures, math.GR / math.GT) that gives the first explicit polynomial upper bound on the Dehn function of a finitely presented non-hyperbolic subg…