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A free group of piecewise linear transformations

Grzegorz Tomkowicz (2011)

Colloquium Mathematicae

We prove the following conjecture of J. Mycielski: There exists a free nonabelian group of piecewise linear, orientation and area preserving transformations which acts on the punctured disk {(x,y) ∈ ℝ²: 0 < x² + y² < 1} without fixed points.

A locally commutative free group acting on the plane

Kenzi Satô (2003)

Fundamenta Mathematicae

The purpose of this paper is to prove the existence of a free subgroup of the group of all affine transformations on the plane with determinant 1 such that the action of the subgroup is locally commutative.

A note on the continuous extensions of injective morphisms between free groups to relatively fre profinite groups.

Thierry Coulbois, Mark Sapir, Pascal Weil (2003)

Publicacions Matemàtiques

Let V be a pseudovariety of finite groups such that free groups are residually V, and let φ: F(A) → F(B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension φ' of φ between the pro-V completions of F(A) and F(B) is also injective. In particular, if V is extension-closed, this is the case if and only if φ(F(A)) and its pro-V closure in F(B) have the same rank. We examine a number of situations where the injectivity of φ' can...

Brunnian links

Paul Gartside, Sina Greenwood (2007)

Fundamenta Mathematicae

A Brunnian link is a set of n linked loops such that every proper sublink is trivial. Simple Brunnian links have a natural algebraic representation. This is used to determine the form, length and number of minimal simple Brunnian links. Braids are used to investigate when two algebraic words represent equivalent simple Brunnian links that differ only in the arrangement of the component loops.

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