 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000000
S 00029947(XX)00000
SOLVABLE GROUPS WITH POLYNOMIAL DEHN FUNCTIONS.
G. N. ARZHANTSEVA AND D. V. OSIN
Abstract. Given a finitely presented group H , finitely generated subgroup B
of H, and a monomorphism/ : B ! H , we obtain an upper bound of the Dehn
function of the corresponding HNNextension G = hH; t j t \Gamma1 Bt = /(B)i
in terms of the Dehn function of H and the distortion of B in G. Using
such a bound, we construct first examples of nonpolycyclic solvable groups
with polynomial Dehn functions. The constructed groups are metabelian and
contain the solvable BaumslagSolitar groups. In particular, this answers a
question posed in [6].
1. Introduction
Recall that the Denh function of a finitely presented group G = hY j Ri is the
smallest function ffi : N ! N with the following property. For any word w over
Y [ Y \Gamma1 representing the identity in G, there exists a van Kampen diagram over
the presentation of G with the number of 2cells at most ffi (jwj), where jwj denotes
the length of w. As is wellknown, the asymptotic behavior of this function is
