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Bounded Depth Ascending HNN Extensions and
$\unicode[STIX]{x1D70B}_{1}$-Semistability at infinity
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 6 / December 2020
- Published online by Cambridge University Press:
- 22 July 2019, pp. 1529-1550
- Print publication:
- December 2020
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- Article
- Export citation
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A well-known conjecture is that all finitely presented groups have semistable fundamental groups at infinity. A class of groups whose members have not been shown to be semistable at infinity is the class
${\mathcal{A}}$ of finitely presented groups that are ascending HNN-extensions with finitely generated base. The class 
${\mathcal{A}}$ naturally partitions into two non-empty subclasses, those that have “bounded” and “unbounded” depth. Using new methods introduced in a companion paper we show those of bounded depth have semistable fundamental group at infinity. Ascending HNN extensions produced by Ol’shanskii–Sapir and Grigorchuk (for other reasons), and once considered potential non-semistable examples are shown to have bounded depth. Finally, we devise a technique for producing explicit examples with unbounded depth. These examples are perhaps the best candidates to date in the search for a group with non-semistable fundamental group at infinity.
