- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:04:59
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
6
2012
4
On the asymptotics of visible elements and homogeneous equations in surface groups
Yago
Antolín
University of Southampton, SOUTHAMPTON, UNITED KINGDOM
Laura
Ciobanu
Université de Neuchâtel, NEUCHÂTEL, SWITZERLAND
Noèlia
Viles
Universidad Autonoma de Barcelona, BELLATERRA, SPAIN
Free groups, surface groups, equations, visible elements, asymptotic behavior
Let $F$ be a group whose abelianization is $\mathbb{Z}^k$, $k\geq 2$. An element of $F$ is called visible if its image in the abelianization is visible, that is, the greatest common divisor of its coordinates is 1. In this paper we compute three types of densities, annular, even and odd spherical, of visible elements in surface groups. We then use our results to show that the probability of a homogeneous equation in a surface group to have solutions is neither 0 nor 1, as the lengths of the right- and left-hand side of the equation go to infinity.
Group theory and generalizations
Computer science
General
619
638
10.4171/GGD/167
http://www.ems-ph.org/doi/10.4171/GGD/167