Some Examples of Nonfinite Diffeomorphism Groups

Some Examples of Wallpaper Groups

In this note we discuss some symmetry groups as nice examples of group extensions. We begin by a short general discussion of symmetry groups. An isometry of R is a distancepreserving bijection of R. For examples of isometries, translations by a vector, rotations about a point, and reflections about a hyperplane are all isometries. Let Isom(R) be the set of all isometries of R. This is a group u...

THE NONFINITE TYPE OF SOME DiS0M

Homomorphisms between Diffeomorphism Groups

For r ≥ 3, p ≥ 2, we classify all actions of the groups Diffc(R) and Diff+(S) by Cdiffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1-manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question...

Diffeomorphism groups of critical regularity

Let M be a circle or a compact interval, and let α “ k ` τ ě 1 be a real number such that k “ tαu. We write Diff`pMq for the group of orientation preserving Ck diffeomorphisms of M whose kth derivatives are Hölder continuous with exponent τ. We prove that there exists a continuum of isomorphism types of finitely generated subgroups G ď Diff`pMq with the property that G admits no injective homom...

Some Examples and Remarks Concerning Groups

We present some examples and remarks which may be helpful to those who are dealing with an abstract algebra or a first semester group theory course. Alternating groups, dihedral groups, and symmetric groups of small orders are treasure troves of elementary examples and counter examples concerning groups. Mathematics Subject Classification: 20-01, 20-02, 20B05, 20B07, 20E06