On pro-isomorphic zeta functions of D∗D^*-groups of even Hirsch length

Pro-isomorphic zeta functions of finitely generated nilpotent groups form one of the group-theoretic generalisations of the Riemann zeta functions. They are Dirichlet generating functions enumerating finite-index subgroups whose profinite completion is isomorphic to that of the ambient group. We study pro-isomorphic zeta functions of $D^*$-groups; these form the building blocks of finitely generated class two nilpotent groups with centre of rank two, up to commensurability. These groups were classified by Grunewald and Segal, and can be indexed by primary polynomials whose companion matrices define commutator relations. We provide a key step towards the elucidation of the pro-isomorphic zeta functions of $D^*$-groups of even Hirsch length by describing the automorphism groups of the associated graded Lie rings. Utilizing this description of the automorphism groups, we calculate the local pro-isomorphic zeta functions of groups associated to the polynomials $x^2$ and $x^3$. In both cases, the local zeta functions are uniform in the prime~$p$ and satisfy functional equations.Comment: 29 page

Uniform cell decomposition with applications to Chevalley groups

We express integrals of definable functions over definable sets uniformly for non-Archimedean local fields, extending results of Pas. We apply this to Chevalley groups, in particular proving that zeta functions counting conjugacy classes in congruence quotients of such groups depend only on the size of the residue field, for sufficiently large residue characteristic. In particular, the number of conjugacy classes in a congruence quotient depends only on the size of the residue field. The same holds for zeta functions counting dimensions of Hecke modules of intertwining operators associated to induced representations of such quotients.Comment: 20 pages, final version, to appear in the Journal of the LM

Extrinsic Rewards and Intrinsic Motives: Standard and Behavioral Approaches to Agency and Labor Markets

Employers structure pay and employment relationships to mitigate agency problems. A large literature in economics documents how the resolution of these problems shapes personnel policies and labor markets. For the most part, the study of agency in employment relationships relies on highly stylized assumptions regarding human motivation, e.g., that employees seek to earn as much money as possible with minimal effort. In this essay, we explore the consequences of introducing behavioral complexity and realism into models of agency within organizations. Specifically, we assess the insights gained by allowing employees to be guided by such motivations as the desire to compare favorably to others, the aspiration to contribute to intrinsically worthwhile goals, and the inclination to reciprocate generosity or exact retribution for perceived wrongs. More provocatively, from the standpoint of standard economics, we also consider the possibility that people are driven, in ways that may be opaque even to themselves, by the desire to earn social esteem or to shape and reinforce identity

Social Relations and Relational Incentives

This paper studies how social relationships between managers and employees affect relational incentive contracts. To this end we develop a simple dynamic principal-agent model where both players may have feelings of altruism or spite toward each other. The contract may contain two types of incentives for the agent to work hard: a bonus and a threat of dismissal. We find that good social relationships undermine the credibility of a threat of dismissal but strengthen the credibility of a bonus. Among others, these two mechanisms imply that better social relationships sometimes lead to higher bonuses, while worse social relationships may increase productivity and players' utility in equilibrium

A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions

The pro-isomorphic zeta function ζ∧Γ(s) of a finitely generated nilpotent group Γ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of Γ . Such zeta functions can be expressed as Euler products of p-adic integrals over the Qp -points of an algebraic automorphism group associated to Γ . In this way they are closely related to classical zeta functions of algebraic groups over local fields. We describe the algebraic automorphism groups for a natural family of class-2 nilpotent groups Δm,n of Hirsch length (m+n−2n−1)+(m+n−1n−1)+n and central Hirsch length n; these groups can be viewed as generalisations of D∗ -groups of odd Hirsch length. General D∗ -groups, that is ‘indecomposable’ finitely generated, torsion-free class-2 nilpotent groups with central Hirsch length 2, were classified up to commensurability by Grunewald and Segal. We calculate the local pro-isomorphic zeta functions for the groups Δm,n and obtain, in particular, explicit formulae for the local pro-isomorphic zeta functions associated to D∗ -groups of odd Hirsch length. From these we deduce local functional equations; for the global zeta functions we describe the abscissae of convergence and find meromorphic continuations. We deduce that the spectrum of abscissae of convergence for pro-isomorphic zeta functions of class-2 nilpotent groups contains infinitely many cluster points. For instance, the global abscissa of convergence of the pro-isomorphic zeta function of a D∗ -group of Hirsch length 2m+3 is shown to be 6−15m+3 .The first author thanks the University of Cape Town and Ort Braude College for travel grants. The second author acknowledges support by DFG grant KL 2162/1-1. The third author acknowledges the support of the Australian Research Council and the Israel Science Foundation (Grant nos. 1862/16 and 382/11)