Finiteness of outer automorphism groups of random right-angled Artin groups

We consider the outer automorphism group Out(A_Gamma) of the right-angled Artin group A_Gamma of a random graph Gamma on n vertices in the Erdos--Renyi model. We show that the functions (log(n)+log(log(n)))/n and 1-(log(n)+log(log(n)))/n bound the range of edge probability functions for which Out(A_Gamma) is finite: if the probability of an edge in Gamma is strictly between these functions as n grows, then asymptotically Out(A_Gamma) is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically Out(A_Gamma) is almost surely infinite. This sharpens results of Ruth Charney and Michael Farber from their preprint "Random groups arising as graph products", arXiv:1006.3378v1.Comment: 29 pages. Mostly rewritten, results tightened, statements corrected, gaps fille

On the second homology group of the Torelli subgroup of Aut(F_n)

Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H_2(IA_n) as a GL_n(Z)-module. Corollaries include a version of surjective representation stability for H_2(IA_n), the vanishing of the GL_n(Z)-coinvariants of H_2(IA_n), and the vanishing of the second rational homology group of the level l congruence subgroup of Aut(F_n). Our generating set is derived from a new group presentation for IA_n which is infinite but which has a simple recursive form.Comment: 39 pages; minor revision; to appear in Geom. Topo

A Birman exact sequence for the Torelli subgroup of Aut(F_n)

We develop an analogue of the Birman exact sequence for the Torelli subgroup of Aut(F_n). This builds on earlier work of the authors who studied an analogue of the Birman exact sequence for the entire group Aut(F_n). These results play an important role in the authors' recent work on the second homology group of the Torelli group.Comment: 31 pages, minor revision; to appear in Int. J. Algebr. Compu

Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we call mapping class groups over graphs, are indexed over labeled simplicial graphs with 2g vertices. The mapping class group over the graph Gamma is defined to be a subgroup of the automorphism group of the right-angled Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a theorem of Magnus.Comment: 45 page

Full-featured peak reduction in right-angled Artin groups

We prove a new version of the classical peak-reduction theorem for automorphisms of free groups in the setting of right-angled Artin groups. We use this peak-reduction theorem to prove two important corollaries about the action of the automorphism group of a right-angled Artin group $A_\Gamma$ on the set of $k$-tuples of conjugacy classes from $A_\Gamma$: orbit membership is decidable, and stabilizers are finitely presentable. Further, we explain procedures for checking orbit membership and building presentations of stabilizers. This improves on a previous result of the author's. We overcome a technical difficulty from the previous work by considering infinite generating sets for the automorphism groups. The method also involves a variation on the Hermite normal form for matrices.Comment: 72 pages, 1 figure. Updated to incorporate referee comment

Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [Algebr. Geom. Topol. 7 (2007):1297-1326]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent Lie algebras. In particular, we develop a notion of the "polynomial straightening" of a singular homology chain in a nilpotent homogeneous space.Comment: 34 page

Quotients of the braid group that are extensions of the symmetric group

We consider normal subgroups $N$ of the braid group $B_n$ such that the quotient $B_n/N$ is an extension of the symmetric group by an abelian group. We show that, if $n\geq 4$, then there are exactly 8 commensurability classes of such subgroups. We define a Specht subgroup to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions $(n-1,1)$ and $(n-2,2)$, working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions.Comment: 44 page