Periodic automorphisms of the two-generator free group

Let N g denote a closed nonorientable surface of genus g. For g ≥ 2 the mapping class group M(N g ) is generated by Dehn twists and one crosscap slide (Y -homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We give necessary and sufficient conditions for the existence of roots of crosscap slides and crosscap transpositions.

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“…Moreover, there is only one conjugacy class of such elements in GL(2, Z)-for details see for example Theorem 2 of [7].…”

Section: The Case Of G =mentioning

confidence: 99%

Roots of crosscap slides and crosscap transpositions

Parlak

Stukow

2017

Period Math Hung

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“…Moreover, any element of Aut(F 2 ) can be written uniquely in the form p r u(x, y)x 2s w(τ a , τ b ) where r, s ∈ {0, 1}, w(τ a , τ b ) is a reduced word in Inn(F 2 ) and u(x, y) is a reduced word where x, y, y −1 are the only powers of x, y appearing (see [11,10]). …”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Then it necessarily conjugates elements of order 4 to elements of order 4. But the only elements of order 4 in Aut(F 2 ) are conjugates of x ±1 (see [11]). Hence we have the following relation…”

Section: Preliminaries and Notationmentioning

confidence: 99%

On the K-theory of certain extensions of free groups

Metaftsis

Prassidis

2015

Forum Mathematicum

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Since $\operatorname*{Hol}(F_{n})$ embeds into $\operatorname*{Aut}(F_{n+1})$, one can construct inductively the subgroups ${\mathcal{H}}_{(n)}$ of $\operatorname*{Aut}(F_{n+1})$ by setting ${{\mathcal{H}}_{(1)}=\operatorname*{Hol}(F_{2})}$ and ${{\mathcal{H}}_{(n)}=F_{n+1}\rtimes{\mathcal{H}}_{(n-1)}}$. We show that the FJCw holds for ${\mathcal{H}}_{(n)}$. Moreover, we calculate the lower K-theory for the groups ${\mathcal{H}}_{(n)}$.

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“…This can be seen easily from the presentation of these groups and the fact that elements of finite order in GL(2, Z) have order 2, 3, 4, or 6. Furthermore, there are only three elements of order 2 and one element of order 3, 4, or 6 up to conjugation [8]. By Lemma 1, these groups cannot contain a normal subgroup of rank 2.…”

mentioning

confidence: 97%

Abelian rank of normal torsion-free finite index subgroups of polyhedral groups

Lee¹

1985

Trans. Amer. Math. Soc.

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Abstract.Suppose that P is a convex polyhedron in the hyperbolic 3-space with finite volume and P has integer ( > 1) submultiples of it as dihedral angles. We prove that if the rank of the abelianization of a normal torsion-free finite index subgroup of the polyhedral group G associated to P is one, then P has exactly one ideal vertex of type (2,2,2,2) and G has an index two subgroup which does not contain any one of the four standard generators of the stabilizer of the ideal vertex.

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Periodic automorphisms of the two-generator free group

Cited by 13 publications

References 0 publications

Roots of crosscap slides and crosscap transpositions

Roots of crosscap slides and crosscap transpositions

On the K-theory of certain extensions of free groups

Abelian rank of normal torsion-free finite index subgroups of polyhedral groups

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