Metabelian Groups With Quadratic Dehn Function and Baumslag–solitar Groups

Abstract

We prove that metabelian locally compact groups in a certain class have quadratic Dehn function. As an application, we embed the solvable Baumslag-Solitar groups in finitely presented metabelian groups with quadratic Dehn function. Also, we prove that Baumslag's finitely presented metabelian groups, in which the lamplighter groups embed, have quadratic Dehn function.

The Dehn function of Baumslag’s metabelian group

“…A polynomial bound on the Dehn functions of m in an earlier version of this article spurred de Cornulier and Tessera to prove in [12] that the Dehn function of p is quadratic for all prime p. They view p , for p prime, as a cocompact lattice in Sol 5 (F p ((t))), which they prove enjoys a quadratic isoperimetric function by adapting Gromov's proof from [18] of the corresponding result for Sol 5 (R). They argue that their methods can be elaborated to cover m for all m.…”

“…These groups appear in other guises: both and m are groups of affine matrices and have Cayley graphs that are horocyclic products of trees [3]; for p prime, p is a cocompact lattice in Sol 5 (F p ((t))) [12].…”

Asymptotic invariants, complexity of groups and related problems

“…Dehn functions (some of these groups contain the Baumslag-Solitar group BS(1, 2)) [59]. (3) The Heisenberg groups of rank ≥ 5, sufficiently large central power of any (infinite) finitely generated nilpotent group of class 2, some finitely generated nilpotent groups of arbitrary nilpotency class [3,185,237,238].…”

“…Of course as we have seen (Example 2.22), the Baumslag-Solitar group has exponential Dehn function. But Druţu [73], and later Cornulier and Tessera [59] have shown that metabelian groups of higher ranks can have quadratic Dehn functions. The key property of Young's solvable subgroups is the freedom of choosing them.…”

“…In some particular cases like the free Burnside groups B m,n , the free solvable groups, etc., it is possible to modify this construction and get simple presentations of groups with polynomial isoperimetric functions where these groups embed [186]. For the Baumslag-Solitar group BS(1, 2) it was done in [186] too, and an even much simpler presentation was found by Cornulier and Tessera [59] (see also Theorem 4.1, part (2)).…”