Some quasi-isometries of solvable groups
by
Lee Mosher
Rutgers University, Newark
Coauthors: Benson Farb

Given a finitely generated group G with the word metric, the set of self quasi-isometries of G, modulo identification of quasi-isometries which differ by a bounded amount, forms a group under composition denoted QI(G), the quasi-isometry group of G. Identifying QI(G) is important in classifying finitely generated groups which are quasi-isometric to G. We have calculated QI(BS(1, n)) where BS(1, n) = <a, b | bab^-1 = aⁿ >, the solvable Baumslag-Solitar group. The result is: QI(BS(1, n)) = Bilip(R) ×Bilip(Q_n), where R is the real line, Q_n is metric space of n-adic rational numbers, and Bilip(X) is the group of bilipschitz homeomorphisms of the real line.

In this talk I will describe some big subgroups of QI(G) for some classes of solvable groups G. For example, if G is a lattice in SOLV, the 3-dimensional solvable Lie group, then QI(G) contains a subgroup of the form (Bilip(R) ×Bilip(R)) \semidirect Z/2, where Z/2 acts by permuting the factors.

Date received: March 11, 1998