Vol. 9, No. 3, 2016

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Strong depth and quasigeodesics in finitely generated groups

Brian Gapinski, Matthew Horak and Tyler Weber

Vol. 9 (2016), No. 3, 367–377

A “dead end” in the Cayley graph of a finitely generated group is an element beyond which no geodesic ray issuing from the identity can be extended. We study the so-called “strong dead-end depth” of group elements and its relationship with the set of infinite quasigeodesic rays issuing from the identity. We show that the ratio of strong depth to word length is bounded above by 1 2 in every finitely generated group and that for any element g in a finitely generated group G, there is an infinite (3,0)-quasigeodesic ray issuing from the identity and passing through g. Applying the Švarc–Milnor lemma to a finitely generated group acting geometrically on a geodesically connected metric space, we obtain the result that for any two points in such a space, there is an infinite quasigeodesic ray starting at one and passing through the other with quasigeodesic constants independent of the points selected.

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Cayley graph, dead end, quasigeodesic
Mathematical Subject Classification 2010
Primary: 20F65
Received: 13 June 2014
Revised: 19 July 2015
Accepted: 22 July 2015
Published: 3 June 2016

Communicated by Kenneth S. Berenhaut
Brian Gapinski
Department of Mathematics and Computer Science
Wesleyan University
Middletown, CT 06459
United States
Matthew Horak
Department of Mathematics, Statistics and Computer Science
University of Wisconsin-Stout
Menomonie, WI 54751
United States
Tyler Weber
Department of Mathematics, Statistics and Computer Science
University of Wisconsin-Stout
Menomonie, WI 54751
United States