An example of a finite presented solvable group

Limits of covering spaces and residual properties of groups

“…Recall the setup of the introduction: a group G and a series of subgroups (1) of G are given; X is a locally 1-connected topological space with K\(X) = G and we form the sequence (2) of covering spaces corresponding to the sequence of subgroups (1). (Assume throughout that all spaces are connected.…”

“…Now, to make a counterexample to Griffiths' claim, start with a finitely presented solvable group G that is not residually finite. For the existence of such groups, see Abels [1]. Construct a series of subgroups (1) satisfying (i)-(iii) as above.…”

“…. (2) corresponding to the series of subgroups (1). Denote the inverse limit of the system (2) of CO topological spaces and maps by lim X n .…”

“…Suppose the quotient G/G n is a ^-group for each term in the series (1). Then if one can show the singular homology H l (\imX,,) = 0, it follows that H l (X J ) = 0; that is, J = it\(Xj) is a 278 JON MICHAEL CORSON perfect group.…”

On the wgsc and qsf tameness conditions for finitely presented groups

“…Thus all solvable groups that are not virtually abelian are not CAT.0/ and many of them are qsf (e.g., if their center is not torsion). Notice that there exist solvable groups with infinitely generated centers, as those constructed by Abels (see [1]). In general we do not know whether all solvable groups (in particular those with finite centers) are qsf, but one can prove that Abels' group is qsf since it is an S-arithmetic group.…”