Normal generators for mapping class groups are abundant

In 1974, Thurston proved that, up to isotopy, every automorphism of closed orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has lead to a rich theory with applications ranging from dynamical systems to low dimensional topology. Associated with every pseudo-Anosov map is a real number $\lambda > 1$ known as the stretch factor. Thurston showed that every stretch factor is an algebraic unit but it is unknown exactly which units can appear as stretch factors. In this paper we show that every Salem number has a power that is the stretch factor of a pseudo-Anosov map arising from a construction due to Thurston. We also show that every totally real number field $K$ is of the form $K = \mathbb{Q}(\lambda + \lambda^{-1})$, where $\lambda$ is the stretch factor of a pseudo-Anosov map arising from Thurston's construction.

Improved routing strategies with succinct tables

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

We show that the Nielsen–Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum $SU(n)$ representations, for any fixed $n\geq 2$. In the Pseudo–Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) $SU(2)$-TQFT representation matrices. It follows that at big enough levels, Pseudo–Anosov mapping classes are represented by matrices of infinite order.

In the theory of cluster algebras, a mutation loop induces discrete dynamical systems via its actions on the cluster \({\mathcal {A}}\)- and \({\mathcal {X}}\)-varieties. In this paper, we introduce a property of mutation loops, called the sign stability, with a focus on the asymptotic behavior of the iteration of the tropical \({\mathcal {X}}\)-transformation. The sign stability can be thought of as a cluster algebraic analogue of the pseudo-Anosov property of a mapping class on a surface. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster \({\mathcal {A}}\)- and \({\mathcal {X}}\)-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This gives a cluster algebraic analogue of the classical theorem which relates the topological entropy of a pseudo-Anosov mapping class with its stretch factor.

Let G be a geometric graph whose vertex set S is a set of n points in ℝ d . The stretch factor of two distinct points p and q in S is the ratio of their shortest-path distance in G and their Euclidean distance. We consider the problem of approximating the average of the n choose 2 stretch factors determined by all pairs of points in S . We show that for paths, cycles, and trees, this average can be approximated, within a factor of 1+e, in O ( n polylog( n )) time. For plane graphs in ℝ 2 , we present a (2+e)-approximation algorithm with running time O ( n 5/3 polylog( n )), and a (4+e)-approximation algorithm with running time O ( n 3/2 polylog( n )). Finally, we show that, for any tree in ℝ 2 , the exact average of the squares of the n choose 2 stretch factors can be computed in O ( n 11/6 ) time.

Two conflicting goals play a crucial role in the design of routing schemes for communication networks. A routing scheme should use paths that are as short as possible for routing messages in the network, while keeping the routing information stored in the processors' local memory as succinct as possible. The efficiency of a routing scheme is measured in terms of its stretch factor -the maximum ratio between the length of a route computed by the scheme and that of a shortest path connecting the same pair of vertices. Most previous work has concentrated on finding good routing schemes (with a small fixed stretch factor) for special classes of network topologies. In this paper the problem for general networks is studied, and the entire range of possible stretch factors is examined. The results exhibit a trade-off between the efficiency of a routing scheme and its space requirements. Almost tight upper and lower bounds for this trade-off are presented. Specifically, it is proved that any routing scheme for general n -vertex networks that achieves a stretch factor k ≥ 1 must use a total of Ω( n 1+1/(2 k +4) ) bits of routing information in the networks. This lower bound is complemented by a family K ( k ) of hierarchical routing schemes (for every k ≥ l) for unit-cost general networks, which guarantee a stretch factor of O ( k ), require storing a total of O ( k 3 n 1+(1/h) log n )- bits of routing information in the network, name the vertices with O (log 2 n )-bit names and use O (log n )-bit headers.

Consider the mapping class group Mod g,p of a surface Σ g,p of genus g with p punctures, and a finite collection {f1, . . . , fk} 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, . . . , fk} 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 .

Given ϕ \phi a pseudo-Anosov map, let ℓ T ( ϕ ) \ell _\mathcal {T}(\phi ) denote the translation length of ϕ \phi in the Teichmüller space, and let ℓ C ( ϕ ) \ell _{\mathcal {C}}(\phi ) denote the stable translation length of ϕ \phi in the curve graph. Gadre–Hironaka–Kent–Leininger showed that, as a function of Euler characteristic χ ( S ) \chi (S) , the minimal possible ratio τ ( ϕ ) = ℓ T ( ϕ ) ℓ C ( ϕ ) \tau (\phi ) = \frac {\ell _\mathcal {T}(\phi )}{\ell _\mathcal {C}(\phi )} is log ⁡ ( | χ ( S ) | ) \log (|\chi (S)|) , up to uniform additive and multiplicative constants. In this short note, we introduce a new construction of such ratio optimizers and demonstrate their abundance in the mapping class group. Further, we show that ratio optimizers can be found arbitrarily deep into the Johnson filtration as well as in the point pushing subgroup.

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from 1, -1, i, -i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations. The unitary representations in the interval [-1,0] interpolate analytically between two natural geometric unitary representations, the SU(2)-character variety representation studied by Goldman and the multicurve representation induced by the action of the mapping class group on multicurves. The finite-dimensional representations converge analytically to the infinite-dimensional ones. We recover Marche and Narimannejad's convergence theorem, and Andersen, Freedman, Walker and Wang's asymptotic faithfulness, that states that the image of a non-central mapping class is always non-trivial after some level r. When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level r in term of its dilatation.

In this note we make progress toward a conjecture of Durham–Fanoni–Vlamis, showing that every infinite-type surface with fi­ni­te-invariance index \(1\) and no nondisplaceable compact subsurfaces fails to have a good graph of curves, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and with a natural mapping class group action having infinite diameter orbits. Our arguments use tools developed by Mann–Rafi in their study of the coarse geometry of big mapping class groups.

We give an algorithm to compute the stable lengths of pseudo-Anosovs on the curve graph, answering a question of Bowditch. We also give a procedure to compute all invariant tight geodesic axes of pseudo-Anosovs. Along the way we show that there are constants 1 &gt; a 1 &gt; a 2 1&gt;a_1&gt;a_2 such that the minimal upper bound on ‘slices’ of tight geodesics is bounded below and above by a 1 ξ ( S ) a_1^{\xi (S)} and a 2 ξ ( S ) a_2^{\xi (S)} , where ξ ( S ) \xi (S) is the complexity of the surface. As a consequence, we give the first computable bounds on the asymptotic dimension of curve graphs and mapping class groups. Our techniques involve a generalization of Masur–Minsky’s tight geodesics and a new class of paths on which their tightening procedure works.

Braids, metallic ratios and periodic solutions of the 2n-body problem

McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $\sigma,\tau \in \mathrm{Mod}(S)$ there is some power $N$ such that the group $\langle \sigma^N,\tau^N\rangle$ is either free of rank two or abelian, and gives a geometric criterion for the dichotomy. The analogous statement is false in linear groups, and unresolved for outer automorphisms of a free group. Several analogs are known for exponentially growing outer automorphisms satisfying various technical hypothesis. In this article we prove an analogous statement when $\sigma$ and $\tau$ are linearly growing outer automorphisms of $F_r$, and give a geometric criterion for the dichotomy. Further, Hamidi-Tehrani proved that for Dehn twists in the mapping class group this independence dichotomy is \emph{uniform}: $N=4$ suffices. In a similar style, we obtain an $N$ that depends only on the rank of the free group.

We show that given a collection \(X=\{f_1\), ..., \(f_m\}\) of pure mapping classes on a surface S, there is an explicit constant N, depending only on X, such that their Nth powers \(\{f_1^N\), ..., \(f_m^N\}\) generate the expected right-angled Artin subgroup of MCG(S). Moreover, we show that these subgroups are undistorted, and that each element is pseudo-Anosov on the largest possible subsurface.