Browsing by Author "Putman, Andrew"
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Item Abelian quotients of subgroups of the mapping class group and higher Prym representations(London Mathematical Society, 2013-08) Putman, Andrew; Wieland, BenA well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto Z if the genus of the surface is large. We prove that if this conjecture holds for some genus, then it also holds for all larger genera. We also prove that if there is a counterexample to this conjecture, then there must be a counterexample of a particularly simple form. We prove these results by relating the conjecture to a family of linear representations of the mapping class group that we call the higher Prym representations. They generalize the classical symplectic representation.Item Automorphisms of nonpositively curved cube complexes, right-angled Artin groups and homology(2017-04-20) Bregman, Corey Joseph; Putman, Andrew; Wolf, MichaelRecently, the geometry of CAT(0) cube complexes featured prominently in Agol’s resolution of two longstanding conjectures of Thurston in low-dimensional topology: the virtually Haken and virtually fibered conjecture for hyperbolic 3-manifolds. A key step of the proof was to show that every hyperbolic 3-manifold group is virtually special, i.e. virtually the fundamental group of a special nonpositively curved (NPC) cube complex. In this thesis, we study algebraic properties of special groups as they relate to the geometry of special cube complexes. In the first part of the thesis, we introduce a positive integer-valued invariant of special cube complexes called the genus, and show that having genus one is equivalent to having free abelian fundamental group. As a corollary, we obtain a new proof of the fact that every special group is either abelian or surjects onto a non-abelian free group. In the second part of the thesis, we turn our attention to automorphisms of NPC cube complexes. We give a criterion on a special cube complex which implies that any automorphism acts non-trivially on first homology, and show that a non- trivial action on homology can always be achieved by passing to covers. We then apply the criterion to provide a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.Item A Birman exact sequence for Aut(Fn)(Elsevier, 2012) Day, Matthew; Putman, AndrewThe 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 key to many of our proofs are several new generalizations of the Johnson homomorphisms.Item Generating the Johnson filtration(Mathematical Sciences Publishers, 2015) Church, Thomas; Putman, AndrewFor k≥1, let J1g(k) be the k th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k≥1, there exists some Gk≥0 such that J1g(k) is generated by elements which are supported on subsurfaces whose genus is at most Gk. We also prove similar theorems for the Johnson filtration of Aut(Fn) and for certain mod-p analogues of the Johnson filtrations of both the mapping class group and of Aut(Fn). The main tools used in the proofs are the related theories of FI–modules (due to the first author with Ellenberg and Farb) and central stability (due to the second author), both of which concern the representation theory of the symmetric groups over Z.Item Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=−1(Springer, 2015) Brendle, Tara; Margalit, Dan; Putman, AndrewWe prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=−1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.Item The Picard group of the moduli space of curves with level structures(2012) Putman, Andrew; National Science Foundation; Duke University PressFor 4 - L and g large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level L structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level L subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level L subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod L symplectic group with coefficients in the adjoint representation.Item The Rational Cohomology of the Mapping Class Group Vanishes in its Virtual Cohomological Dimension(Oxford University Press, 2012) Church, Thomas; Farb, Benson; Putman, AndrewItem The second rational homology group of the moduli space of curves with level structures(2012) Putman, Andrew; ElsevierLet Γ be a finite-index subgroup of the mapping class group of a closed genus g surface that contains the Torelli group. For instance, Γ can be the level L subgroup or the spin mapping class group. We show that H2(Γ;Q) ∼= Q for g≥5. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to Q. We also prove analogous results for surface with punctures and boundary components.Item Small generating sets for the Torelli group(Mathematical Sciences Publisher, 2012) Putman, AndrewProving a conjecture of Dennis Johnson, we show that the Torelli subgroup Ig of the genus g mapping class group has a finite generating set whose size grows cubically with respect to g. Our main tool is a new space called the handle graph on which Ig acts cocompactly.Item The complex of partial bases for Fn and nite generation of the Torelli subgroup of Aut(Fn)(Springer, 2013) Day, Matthew; Putman, AndrewWe study the complex of partial bases of a free group, which is an analogue for Aut(Fn) of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of Aut(Fn) is highly connected. Using these results, we give a new, topological proof of a theorem of Magnus that asserts that the Torelli subgroup of Aut(Fn) is nitely generated.Item The untwisting number of a knot(2016-04-14) Ince, Kenan A; Putman, AndrewThe unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. First, we show that the algebraic untwisting number is equal to the algebraic unknotting number. However, we also exhibit several families of knots for which the difference between the unknotting and untwisting numbers is arbitrarily large, even when we only allow twists on a fixed number of strands or fewer. Second, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to show that several 10-and 11-crossing knots cannot be unknotted by a single positive or negative generalized crossing change. We also use the Ozsváth-Szabó tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p and q.