Automorphisms of graph products of groups from a geometric perspective

This article studies automorphism groups of graph products of arbitrary groups. We completely characterise automorphisms that preserve the set of conjugacy classes of vertex groups as those automorphisms that can be decomposed as a product of certain elementary automorphisms (inner automorphisms, partial conjugations, automorphisms associated to symmetries of the underlying graph). This allows us to completely compute the automorphism group of certain graph products, for instance in the case where the underlying graph is finite, connected, leafless and of girth at least $5$. If in addition the underlying graph does not contain separating stars, we can understand the geometry of the automorphism groups of such graph products of groups further: we show that such automorphism groups do not satisfy Kazhdan's property (T) and are acylindrically hyperbolic. Applications to automorphism groups of graph products of finite groups are also included. The approach in this article is geometric and relies on the action of graph products of groups on certain complexes with a particularly rich combinatorial geometry. The first such complex is a particular Cayley graph of the graph product that has a quasi-median geometry, a combinatorial geometry reminiscent of (but more general than) CAT(0) cube complexes. The second (strongly related) complex used is the Davis complex of the graph product, a CAT(0) cube complex that also has a structure of right-angled building.Comment: 36 pages. The article subsumes and vastly generalises our preprint arXiv:1803.07536. To appear in Proc. Lond. Math. So

The Deviance of the Zookeepers

In May 1968 Alvin Gouldner published his attack on the ‘Becker School’ of sociology (‘The Sociologist as Partisan’). The essay was a sometimes sarcastic and brutal but characteristically insightful and sharp critique of what he called the ‘Becker School’ of sociology – especially as it related to law-breaking and norm-transgressing outsiders. In attacking the failure of ‘sceptical deviancy theory’ to confront the wider structural sources of power and authority, its seeming inability to address gross social divisions of wealth and status, and its lack of attention to the larger political and economic interests that were embedded in departments of State and industrial and financial corporations alike, Gouldner pinpointed with some accuracy the radical motivations of the soon-to-emerge ‘new criminology’ – in both its ‘left idealist’ and ‘left realist’ guises. What Gouldner’s essay really exposed was a certain kind of ‘deviant imagination’ (c.f., Pearson, 1975) prevalent in the emerging critical criminologies of 1960s America (and then the UK, see Young, 1969). In this paper I use Gouldner’s essay as a lens to investigate the ‘deviant imagination’ of contemporary critical criminologies and ask: who are the zookeepers of contemporary criminology and what is their deviant imagination

Social Security Wealth: The Impact of Alternative Inflation Adjustments

The distribution of wealth is one of the most important and least studied features of our economic life. A lack of good data on household wealth is the primary reason for the inadequate attention to this subject. Moreover, the evidence that is available from household surveys and estate records excludes the most important asset of the vast majority of households: the value of future social security benefits. The purpose of the current paper is to present evidence on the distribution of social security wealth and to use these estimates to analyze the impact of alternative methods of adjusting future benefits for changes in the price level.

Quantum random number generation on a mobile phone

Quantum random number generators (QRNGs) can significantly improve the security of cryptographic protocols, by ensuring that generated keys cannot be predicted. However, the cost, size, and power requirements of current QRNGs has prevented them from becoming widespread. In the meantime, the quality of the cameras integrated in mobile telephones has improved significantly, so that now they are sensitive to light at the few-photon level. We demonstrate how these can be used to generate random numbers of a quantum origin

Quadratization of Symmetric Pseudo-Boolean Functions

A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to $\mathbb{R}$. For a pseudo-Boolean function $f(x)$ on $\{0,1\}^n$, we say that $g(x,y)$ is a quadratization of $f$ if $g(x,y)$ is a quadratic polynomial depending on $x$ and on $m$ auxiliary binary variables $y_1,y_2,\ldots,y_m$ such that $f(x)= \min \{g(x,y) : y \in \{0,1\}^m \}$ for all $x \in \{0,1\}^n$. By means of quadratizations, minimization of $f$ is reduced to minimization (over its extended set of variables) of the quadratic function $g(x,y)$. This is of some practical interest because minimization of quadratic functions has been thoroughly studied for the last few decades, and much progress has been made in solving such problems exactly or heuristically. A related paper \cite{ABCG} initiated a systematic study of the minimum number of auxiliary $y$-variables required in a quadratization of an arbitrary function $f$ (a natural question, since the complexity of minimizing the quadratic function $g(x,y)$ depends, among other factors, on the number of binary variables). In this paper, we determine more precisely the number of auxiliary variables required by quadratizations of symmetric pseudo-Boolean functions $f(x)$, those functions whose value depends only on the Hamming weight of the input $x$ (the number of variables equal to $1$).Comment: 17 page

Gravitational waves and electroweak baryogenesis in a global study of the extended scalar singlet model

We perform a global fit of the extended scalar singlet model with a fermionic dark matter (DM) candidate. Using the most up-to-date results from the $\mathit{Planck}$ measured DM relic density, direct detection limits from the XENON1T (2018) experiment, electroweak precision observables and Higgs searches at colliders, we constrain the 7-dimensional model parameter space. We also find regions in the model parameter space where a successful electroweak baryogenesis (EWBG) can be viable. This allows us to compute the gravitational wave (GW) signals arising from the phase transition, and discuss the potential discovery prospects of the model at current and future GW experiments. Our global fit places a strong upper $\mathit{and}$ lower limit on the second scalar mass, the fermion DM mass and the scalar-fermion DM coupling. In agreement with previous studies, we find that our model can simultaneously yield a strong first-order phase transition and saturate the observed DM abundance. More importantly, the GW spectra of viable points can often be within reach of future GW experiments such as LISA, DECIGO and BBO.Comment: 42 pages, 10 figures and 2 tables; v2: updated references, submitted to JHEP; v3: corrected typos and updated references, matches version published in JHE