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Benson Farb

Publications and source records attributed to Benson Farb.

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

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arXiv2019

Topology and arithmetic of resultants, I: spaces of rational maps

We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree $d$ rational curves in $\mathbb{P}^n$. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen--Cohen--Mann--Milgram, Vassiliev and others.

math.AG↗
arXiv2018

Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.

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arXiv2016

Finite covers of graphs, their primitive homology, and representation theory

Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em primitive homology} group $H_1^{\mathrm{prim}}(Y;\mathbb{C})\subseteq H_1(Y;\mathbb{C})$, which is the span of homology classes represented by components of lifts of primitive elements of $π_1(X)$. This circle of ideas relates combinatorial group theory, surface topology, and representation theory.

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arXiv2016

A vanishing theorem for the homology of discrete subgroups of $\mathrm{Sp}(n,1)$ and $\mathrm{F}_4^{-20}$

For any discrete, torsion-free subgroup $Γ$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(Γ;V)=0$ (resp.\ $H_i(Γ;V)=0$ for $i=13,14,15$) for any $Γ$--module $V$. The main technical advance is a new bound on the $p$--Jacobian of the barycenter map of Besson--Courtois--Gallot. We also apply this estimate to obtain an inequality between the critical exponent and homological dimension of $Γ$, improving on work of M.~Kapovich.

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arXiv2015

Étale homological stability and arithmetic statistics

We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over $\Cb$ in the example of configuration spaces of $n$ points in smooth varieties. To do this, we import the method of homological stability from the realm of topology into the theory of étale cohomology; in particular we give the first examples of stability of étale cohomology groups as Galois representations where the Galois actions are not already explicitly known. We then establish subexponential bounds on the growth of the unstable cohomology, and we apply this and étale homological stability to compute the large $n$ limits of various arithmetic statistics of configuration spaces of varieties over $\F_q$.

math.AG↗
arXiv2015

Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)

We consider the moduli space $\mathcal{R}_n$ of pairs of monic, degree $n$ polynomials whose resultant equals $1$. We relate the topology of these algebraic varieties to their geometry and arithmetic. In particular, we compute their étale cohomology, the associated eigenvalues of Frobenius, and the cardinality of their set of $\mathbb{F}_q$-points. When $q$ and $n$ are coprime, we show that the étale cohomology of $\mathcal{R}_{n/\bar{\mathbb{F}}_q}$ is pure, and of Tate type if and only if $q\equiv 1$ mod $n$. We also deduce the values of these invariants for the finite field counterparts of the moduli spaces $\mathcal{M}_n$ of $SU(2)$ monopoles of charge $n$ in $\mathbb{R}^3$, and the associated moduli space $X_n$ of strongly centered monopoles. An appendix by Cazanave gives an alternative and elementary computation of the point counts.

math.AG↗
arXiv2015

Moving homology classes in finite covers of graphs

Let $Y\to X$ be a finite normal cover of a wedge of $n\geq 3$ circles. We prove that for any $v\neq 0\in H_1(Y;\mathbb{Q})$ there exists a lift $\widetilde{F}$ to $Y$ of a homotopy equivalence $F:X\to X$ so that the set of iterates $\{\widetilde{F}^d(v): d\in \mathbb{Z}\}\subseteq H_1(Y;\mathbb{Q})$ is infinite. The main achievement of this paper is the use of representation theory to prove the existence of a purely topological object that seems to be inaccessible via topology.

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arXiv2014

FI-modules and stability for representations of symmetric groups

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.

math.RT↗
arXiv2014

Torelli buildings and their automorphisms

In this paper we introduce, for each closed orientable surface, an analogue of Tits buildings adjusted to investigation of the Torelli group of this surface. It is a simplicial complex with some additional structure. We call this complex with its additional structure the Torelli building of the surface in question. The main result of this paper shows that Torelli buildings of surfaces of genus at least 5 have only obvious automorphisms, and identifies its group of automorphisms. Namely, we prove that for such a surface every automorphism of its Torelli building is induced by a diffeomorphism of the surface. This theorem about automorphisms of Torelli buildings is intended for applications to automorphisms and virtual automorphisms of Torelli groups. The latter results will be presented on some other occasion. All these results were announced in arXiv:math/0311123.

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arXiv2014

Teichmuller geometry of moduli space, II: M(S) seen from far away

We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the "tangent cone at infinity" of M(S): it is the topological cone on the quotient of the complex of curves C(S) by the mapping class group of S, endowed with an explicitly described metric. The main ingredient is Minsky's product regions theorem.

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arXiv2014

Representation Stability

Representation stability is a phenomenon whereby the structure of certain sequences $X_n$ of spaces can be seen to stabilize when viewed through the lens of representation theory. In this paper I describe this phenomenon and sketch a framework, the theory of FI-modules, that explains the mechanism behind it.

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arXiv2014

FI-modules over Noetherian rings

FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many of the results of [CEF] to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman's central stability for homology of congruence subgroups.

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arXiv2013

Representation stability in cohomology and asymptotics for families of varieties over finite fields

We consider two families X_n of varieties on which the symmetric group S_n acts: the configuration space of n points in C and the space of n linearly independent lines in C^n. Given an irreducible S_n-representation V, one can ask how the multiplicity of V in the cohomology groups H*(X_n;Q) varies with n. We explain how the Grothendieck-Lefschetz Fixed Point Theorem converts a formula for this multiplicity to a formula for the number of polynomials over F_q (or maximal tori in GL_n(F_q), respectively) with specified properties related to V. In particular, we explain how representation stability in cohomology, in the sense of [CF, arXiv:1008.1368] and [CEF, arXiv:1204.4533], corresponds to asymptotic stability of various point counts as n goes to infinity.

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arXiv2013

Representation theory and homological stability

We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood--Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+1)^(n-1) conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied in [CEF] to counting problems in number theory and finite group theory. Representation stability is also used in [C] to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds.

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