Twisting Out Fully Irreducible Automorphisms

By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group.     

On the commutator length of a Dehn twist

We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.     

Crosscap slides and the level 2 mapping class group of a nonorientable surface

Crosscap slide is a homeomorphism of a nonorientable surface of genus at least 2, which was introduced under the name Y-homeomorphism by Lickorish as an example of an element of the mapping class group which cannot be expressed as a product of Dehn twists. We prove that the subgroup of the mapping class group of a closed nonorientable surface N generated by all crosscap slides is equal to the level 2 subgroup consisting of those mapping classes which act trivially on ${H_1(N;\mathbb{Z}_2)}$ . We also prove that this subgroup is generated by involutions.     

一类曲面映射类的分解

设S为至少有一个穿孔点α的Riemann曲面.对于曲线α■S,可以定义关于α■S的Dehn twist t_α.设H是S的映射类群的子群,H中的元素保持α不动,并且投影为S=S∪{α}上平凡的映射类变换.定义t_α是关于α■S的Dehn twist.本文考虑关于X(S上的映射类变换)的方程(t_α■θ)~n■X=圮,其中θ∈H是任意给定的.由于(t_αoθ)~n和t_α~n都投影为关于简单闭曲线■的Dehn twist t_■.所以上述方程在H中的解是存在的.对充分大的n,我们给出上述方程有形如X=θ~(n)的解的充要条件.此外,对任给的θ∈H,刻画了子空间H′■H,这里方程的解X=X_n最终要属于H′.最后,考虑简单映射类变换的某些复合映射,并给出了相应的刻画:它们在沿S上的某些简单曲线做剖分后所得的穿孔pant上是不可约的.     

Degree of roots of disk twists on 3-dimensional handlebodies

Margalit and Schleimer (Geom Topol 13(3):1495–1497, 2009) discovered a nontrivial root of the Dehn twist about a nonseparating curve on a closed oriented connected surface. McCullough and Rajeevsarathy (Geom Dedicata 151(1):397–409, 2011) and Monden (Rocky Mt J Math, to appear) obtained the evaluation of the degrees of roots of Dehn twists. In this paper, we discuss existence and degrees of homeomorphisms whose power is equal to disk twist about a nonseparating disk in the mapping class group of the 3-dimensional handlebody.     

Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups

Consider the mapping class group Mod _g,p of a surface ?? _g,p of genus g with p punctures, and a finite collection {f₁, . . . , f_k} of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N, the mapping classes ${\{f_1^N,\ldots,f_k^N\}}$ generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When {f₁, . . . , f_k} are arbitrary mapping classes, we show that sufficiently large powers of these mapping classes generate a group which embeds in a right-angled Artin group in a controlled way. We establish some analogous results for real and complex hyperbolic manifolds. We also discuss the unsolvability of the isomorphism problem for finitely generated subgroups of Mod _g,p , and recover the fact that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Mod _g,p .     

On Dehn Functions of Infinite Presentations of Groups

We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we obtain a characterization of finitely generated groups for which the word problem can be solved in nondeterministic polynomial time. We also give upper bounds for these functions, as well as for the standard Dehn function, for two well-known periodic groups. In particular, we prove that the (standard) Dehn function of a 2-group Γ of intermediate growth, defined by a system of defining relators due to Lysenok, is bounded from above by C₁x² log₂ x, where C₁ > 1 is a constant. We also show that the (standard) Dehn function of a free m-generator Burnside group B(m, n) of exponent n ≥ 2⁴⁸, where n is either odd or divisible by 2⁹, defined by a minimal system of defining relators, is bounded from above by the subquadratic function x^19/12. Received: September 2007, Revision: March 2008, Accepted: March 2008     

The conjugacy problem for Dehn twist automorphisms of free groups

A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of . Received: February 12, 1996.     

Roots of crosscap slides and crosscap transpositions

Periodica Mathematica Hungarica - Let $$N_{g}$$ denote a closed nonorientable surface of genus g. For $$g \ge 2$$ the mapping class group $$\mathcal {M}(N_{g})$$ is generated by Dehn twists and one...     

The Mapping Class Group of a Nonorientable Surface is Generated by Three Elements and by Four Involutions

We prove that the mapping class group of a closed nonorientable surface is generated by three elements and by four involutions Mathematics Subject Classification (2000). primary: 57N05; secondary: 20F38, 20F05.     

Generalized Markoff maps and McShane's identity

We study the (relative) SL(2,C) character varieties of the one-holed torus and the action of the mapping class group on the (relative) character variety. We show that the subset of characters satisfying two simple conditions called the Bowditch Q-conditions is open in the relative character variety and that the mapping class group acts properly discontinuously on this subset. Furthermore, this is the largest open subset for which this holds. We also show that a generalization of McShane's identity holds for all characters satisfying the Bowditch Q-conditions. Finally, we show that further variations of the McShane-Bowditch identity hold for characters which are fixed by an Anosov element of the mapping class group and which satisfy a relative version of the Bowditch Q-conditions, with applications to identities for incomplete hyperbolic structures on punctured torus bundles over the circle, and also for closed hyperbolic 3-manifolds which are obtained by hyperbolic Dehn surgery on such manifolds.     

Fractional Dehn twists and modular invariants

In this paper, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of pseudo-periodic maps with nontrivial fractional Dehn twist coefficients. We also obtain some uniform lower bounds of non-zero fractional Dehn twist coefficients.     

On the generalized $$L_q$$ L q -reflector inverse problem with variational methods

Geometriae Dedicata - We prove that the mapping class group $$\mathcal {M}(N_g)$$ of a closed nonorientable surface of genus g different than 4 is generated by three torsion elements. Moreover, for...     

The Dehn function of Baumslag’s metabelian group

Baumslag??s group is a finitely presented metabelian group with a ${\mathbb Z \wr \mathbb Z}$ subgroup. There is an analogue with an additional torsion relation in which this subgroup becomes ${C_m \wr \mathbb Z}$ . We prove that Baumslag??s group has an exponential Dehn function. This contrasts with the torsion analogues which have quadratic Dehn functions.     

On upper bounds on stable commutator lengths in mapping class groups

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating curve on an oriented closed surface of genus 2 are not equal to each other.     

On the signature of a Lefschetz fibration coming from an involution

In this article we show that the signature of a Lefschetz fibration coming from a special involution as a product of right-handed Dehn twists depends only on the number of genus on the involution axis. We investigate the geography of such Lefschetz fibrations and we identify it with a blow up of a ruled surface. We also get a geography of the Lefschetz fibration coming from a finite order element of mapping class group as a composition of two special involutions.     

Finitely generated lattices with <Emphasis Type="Italic">M</Emphasis>-standard elements among generators

We prove that a lattice is modular if it is generated by three elements, two of which are M-standard. We also show that a lattice generated by n, n > 3, M-standard elements should not necessarily be modular.     

A Birman exact sequence for Aut(Fn)

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles. We also determine the abelianization of our kernel and build a simple infinite presentation for it. The keys to many of our proofs are several new generalizations of the Johnson homomorphisms.     

Adiabatic limits of η-invariants and the Meyer functions

We construct a function on the orbifold fundamental group of the moduli space of smooth theta divisors, which we call the Meyer function for smooth theta divisors. In the construction, we use the adiabatic limits of the η-invariants of the mapping torus of theta divisors. We shall prove that the Meyer function for smooth theta divisors cobounds the signature cocycle, and we determine the values of the Meyer function for the Dehn twists. In particular, we give an analytic construction of the Meyer function of genus two.