scholarly journals Asymptotic Density of Test Elements in Free Groups and Surface Groups

2016 ◽  
pp. rnw175 ◽  
Author(s):  
Ilir Snopce ◽  
Slobodan Tanushevski
Keyword(s):  
Free Groups ◽  
Asymptotic Density ◽  
Surface Groups ◽  
Test Elements

TEST ELEMENTS IN DIRECT PRODUCTS OF GROUPS

2000 ◽  
Vol 10 (06) ◽  
pp. 751-756 ◽  
Author(s):  
JOHN C. O'NEILL ◽  
EDWARD C. TURNER
Keyword(s):  
Direct Product ◽  
Commutator Subgroup ◽  
Free Groups ◽  
Surface Groups ◽  
Direct Products ◽  
Products Of Groups ◽  
Test Elements ◽  
Torsion Free

We characterize test elements in the commutator subgroup of a direct product of certain groups in terms of test elements of the factors. This provides explicit examples of test elements in direct products whose factors are free groups or surface groups and a tool for doing the same for torsion free hyperbolic factors.

Test Elements, Generic Elements and Almost Primitivity in Free Products

2016 ◽  
Vol 23 (02) ◽  
pp. 263-280
Author(s):  
Ann-Kristin Engel ◽  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Free Groups ◽  
Free Products ◽  
Cyclic Groups ◽  
Test Elements ◽  
Generic Elements

In [5, 6] the relationships between test words, generic elements, almost primitivity and tame almost primitivity were examined in free groups. In this paper we extend the concepts and connections to general free products and in particular to free products of cyclic groups.

scholarly journals Densities in free groups and $\mathbb{Z}^k$, Visible Points and Test Elements

2007 ◽  
Vol 14 (2) ◽  
pp. 263-284 ◽  
Author(s):  
Ilya Kapovich ◽  
Igor Rivin ◽  
Paul Schupp ◽  
Vladimir Shpilrain
Keyword(s):  
Free Groups ◽  
Visible Points ◽  
Test Elements

scholarly journals CANNON–THURSTON MAPS FOR KLEINIAN GROUPS

2017 ◽  
Vol 5 ◽  
Author(s):  
MAHAN MJ
Keyword(s):  
Free Groups ◽  
Kleinian Groups ◽  
Surface Groups ◽  
Finitely Generated ◽  
Ending Lamination

We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

On Some Finiteness Properties in Infinite Groups

2008 ◽  
Vol 15 (01) ◽  
pp. 1-22 ◽  
Author(s):  
Gilbert Baumslag ◽  
Oleg Bogopolski ◽  
Benjamin Fine ◽  
Anthony Gaglione ◽  
Gerhard Rosenberger ◽  
...  
Keyword(s):  
Finite Groups ◽  
Elementary Proof ◽  
Free Groups ◽  
Basic Result ◽  
Surface Groups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Hall's Theorem ◽  
Finiteness Properties ◽  
Hall’S Theorem

We consider some questions concerning finiteness properties in infinite groups which are related to Marshall Hall's theorem. We call these properties Property S and Property R, and they are trivially true in finite groups. We give several elementary proofs using these properties for results on finitely generated subgroups of free groups as well as a new elementary proof of Hall's basic result. Finally, we consider these properties within surface groups and prove an analog of Hall's theorem in that context.

scholarly journals MULTIPLICATIVE MEASURES ON FREE GROUPS

2003 ◽  
Vol 13 (06) ◽  
pp. 705-731 ◽  
Author(s):  
ALEXANDRE V. BOROVIK ◽  
ALEXEI G. MYASNIKOV ◽  
VLADIMIR N. REMESLENNIKOV
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Asymptotic Density ◽  
Density Estimates ◽  
Linear Approximations ◽  
Analytical Properties

We introduce a family of multiplicative distributions {μs|s∈(0,1)} on a free group F and study it as a whole. In this approach, the measure of a given set R⊆F is a function μ(R) : s → μs(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.

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