#### 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
##### Abstract

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 $\frac{1}{2}$ in every finitely generated group and that for any element $g$ in a finitely generated group $G$, there is an infinite $\left(3,0\right)$-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.

##### Keywords
Cayley graph, dead end, quasigeodesic
Primary: 20F65
##### Milestones
Received: 13 June 2014
Revised: 19 July 2015
Accepted: 22 July 2015
Published: 3 June 2016

Communicated by Kenneth S. Berenhaut
##### Authors
 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