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Andrew Putman

Publications and source records attributed to Andrew Putman.

At least 37 records · Page 2Linked to original sources
arXiv2021

The dualizing module and top-dimensional cohomology group of $\text{GL}_n(\mathcal{O})$

For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast to $\text{SL}_n(\mathcal{O})$, we prove that the dualizing module of $\text{GL}_n(\mathcal{O})$ is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of $\text{GL}_n(\mathcal{O})$ in its virtual cohomological dimension.

math.NT↗
arXiv2021

On finite generation of the Johnson filtrations

We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a linear stable range. This was originally proved for the second terms by Ershov and He.

math.GR↗
arXiv2020

Irreducible Sp-representations and subgroup distortion in the mapping class group

We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(Σ;\mathbb{Z})$.

math.GT↗
arXiv2020

On the top dimensional cohomology groups of congruence subgroups of $\text{SL}_n(\mathbb{Z})$

Let $Γ_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of $Γ_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let $\mathcal{T}_n(\mathbb{Q})$ denote the Tits building of $\text{SL}_n(\mathbb{Q})$. Lee-Szczarba conjectured that $H^{\binom{n}{2}}(Γ_n(p))$ is isomorphic to $\widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/Γ_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\binom{n}{2}}(Γ_n(p)) \rightarrow \widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/Γ_n(p))$ is always surjective, but is only injective for $p \leq 5$. In particular, we completely calculate $H^{\binom{n}{2}}(Γ_n(5))$ and improve known lower bounds for the ranks of $H^{\binom{n}{2}}(Γ_n(p))$ for $p \geq 5$.

math.NT↗
arXiv2019

Stability in the homology of unipotent groups

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with $n$ from the point of view of representation stability. Our main theorem asserts that if for each $n$ we have representations $M_n$ of $U_n(R)$ over a ring $\mathbf{k}$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $[n] \mapsto \widetilde{H}_i(U_n(R),M_n)$ defines a finitely generated OI-module. As a consequence, if $\mathbf{k}$ is a field then $dim \widetilde{H}_i(U_n(R),\mathbf{k})$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of $GL_n(\mathcal{O})$ for number rings $\mathcal{O}$.

math.AT↗
arXiv2019

Simple closed curves, finite covers of surfaces, and power subgroups of Out(F_n)

We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set $\mathcal{O} \subset F_n$ which is contained in the union of finitely many $Aut(F_n)$-orbits, we construct finite-index normal subgroups of $F_n$ whose first rational homology is not spanned by powers of elements of $\mathcal{O}$. These examples answer questions of Farb-Hensel, Kent, Looijenga, and Marche. We also show that the quotient of $Out(F_n)$ by the subgroup generated by kth powers of transvections often contains infinite order elements, strengthening a result of Bridson-Vogtmann saying that it is often infinite. Finally, for any set $\mathcal{O} \subset F_n$ which is contained in the union of finitely many $Aut(F_n)$-orbits, we construct integral linear representations of free groups that have infinite image and map all elements of $\mathcal{O}$ to torsion elements.

math.GT↗
arXiv2019

Surface groups, infinite generating sets, and stable commutator length

We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

math.GT↗
arXiv2019

Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)

We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this integrality by proving that the complex of partial bases of O^n is Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results for H^{vcd}(SL_n(O_K); Q), where O_K is the ring of integers in a number field and vcd is the virtual cohomological dimension of SL_n(O_K). The (non)vanishing depends on the (non)triviality of the class group of O_K. We also obtain a vanishing theorem for the cohomology H^{vcd}(SL_n(O_K); V) with twisted coefficients V.

math.NT↗
arXiv2017

Homological vanishing for the Steinberg representation

For a field $k$, we prove that the $i$th homology of the groups $GL_n(k)$, $SL_n(k)$, $Sp_{2n}(k)$, $SO_{n,n}(k)$, and $SO_{n,n+1}(k)$ with coefficients in their Steinberg representations vanish for $n \geq 2i+2$.

math.AT↗
arXiv2017

Generating the Johnson filtration II: finite generation

We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group are finitely generated in a stable range. This was originally proved for the commutator subgroup by Ershov-He.

math.GT↗
arXiv2017

Representation stability and finite linear groups

We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.

math.AT↗
arXiv2016

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.

math.GT↗
arXiv2016

The codimension-one cohomology of SL_n Z

We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n Z, and similarly for GL_n Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GL_n. These theorems are derived from a presentation of the Steinberg module for SL_n Z whose generators are integral apartment classes, generalizing Manin's presentation for the Steinberg module of SL_2 Z. This presentation was originally constructed by Bykovskii. We give a new topological proof of it.

math.NT↗
arXiv2016

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.

math.GT↗
arXiv2015

The Johnson homomorphism and its kernel

We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Johnson's calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.

math.GT↗