Dehn function

Remarks

For functions $f, g \colon \N \to \N$, write $f \preceq g$ if there is $C \ge 1$ with $f(n) \le C g(Cn + C) + Cn + C$ for all $n$, and $f \simeq g$ if $f \preceq g \preceq f$. The Dehn functions of two finite presentations of the same group are $\simeq$-equivalent, so one speaks of the Dehn function of $G$.

Example: for $\Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$, the word $a^n b^n a^{-n} b^{-n}$ of length $4n$ has area $n^2$, and every null-homotopic word of length $n$ has area at most a constant times $n^2$; so $\delta(n) \simeq n^2$.

Example: free groups and finite groups have Dehn function $\simeq n$.

Non-example: the lamplighter group is finitely generated but not finitely presented, so it has no Dehn function in this sense.

A finitely presented group has a solvable word problem if and only if its Dehn function is bounded above by a recursive function. The Baumslag–Solitar group $\langle a, b \mid b a b^{-1} = a^2 \rangle$ has an exponential Dehn function.