Exponential Radicals of Solvable Lie Groups

AbstractFor any connected Lie group G, we introduce the notion of exponential radical Exp(G) that is the set of all strictly exponentially distorted elements of G. In case G is a connected simply-connected solvable Lie group, we prove that Exp(G) is a connected normal Lie subgroup in G and the exponential radical of the quotient group G/Exp(G) is trivial. Using this result, we show that the relative growth function of any subgroup in a polycyclic group is either polynomial or exponential

Peripheral fillings of relatively hyperbolic groups

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $\pi_1(M)$. The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and the group $G$ is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Various applications of these results are discussed.Comment: The difference with the previous version is that Proposition 3.2 is proved for quasi--geodesics instead of geodesics. This allows to simplify the exposition in the last section. To appear in Invent. Mat

Solvable groups with polynomial Dehn functions

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 HNN-extension G = hH; t j t1Bt = (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 non-polycyclic solvable groups with polynomial Dehn functions. The constructed groups are metabelian and contain the solvable Baumslag-Solitar groups. In particular, this answers a question posed by Birget, Ol'shanskii, Rips, and Sapir