Southampton

The Haagerup property is a group theoretic property which is a strong converse of Kazhdan's property (T). It implies the Baum-Connes conjecture and has connections with amenability, C*-algebras, representation theory and so on. It is thus not surprising that quite some effort was made to investigate how the Haagerup property behaves under the formation of free products, direct products, direct limits,... In joint work with Y.Antolin, we investigated the behaviour of the Haagerup property under graph products. In this talk we introduce the concept of a graph product, we give a gentle introduction to the Haagerup property and we discuss its behaviour under graph products.

I will introduce the notion of Kurosh rank for subgroups of 
free products. This rank satisfies the Howson property, i.e. the 
intersection of two subgroups of finite Kurosh rank has finite Kurosh rank.
I will present a version of the Strengthened Hanna Neumann inequality in 
the case of free products of right-orderable groups. Joint work with  A. 
Martino and I. Schwabrow.

Self-renewal and pluripotency of mouse embryonic stem (ES) cells are controlled by a complex transcriptional regulatory network (TRN) which is rich in positive feedback loops. A number of key components of this TRN, including Nanog, show strong temporal expression fluctuations at the single cell level, although the precise molecular basis for this variability remains unknown. In this talk I will discuss recent work which uses a genetic complementation strategy to investigate genome-wide mRNA expression changes during transient periods of Nanog down-regulation. Nanog removal triggers widespread changes in gene expression in ES cells. However, we found that significant early changes in gene expression were reversible upon re-induction of Nanog, indicating that ES cells initially adopt a flexible “primed” state. Nevertheless, these changes rapidly become consolidated irreversible fate decisions in the continued absence of Nanog. Using high-throughput single cell transcriptional profiling we observed that the early molecular changes are both stochastic and reversible at the single cell level. Since positive feedback commonly gives rise to phenotypic variability, we also sought to determine the role of feedback in regulating ES cell heterogeneity and commitment. Analysis of the structure of the ES cell TRN revealed that Nanog acts as a feedback “linchpin”: in its presence positive feedback loops are active and the extended TRN is self-sustaining; while in its absence feedback loops are weakened, the extended TRN is no longer self-sustaining and pluripotency is gradually lost until a critical “point-of-no-return” is reached. Consequently, fluctuations in Nanog expression levels transiently activate different sub-networks in the ES cell TRN, driving transitions between a (Nanog expressing) feedback-rich, robust, self-perpetuating pluripotent state and a (Nanog-diminished), feedback-depleted, differentiation-sensitive state. Taken together, our results indicate that Nanog- dependent feedback loops play a central role in controlling both early fate decisions at the single cell level and cell-cell variability in ES cell populations.

The Eilenberg-Ganea conjecture is the statement that every group of cohomological dimension two admits a two-dimensional classifying space.  This problem is unsolved after 50 years.  I shall discuss the background to this question and negative answers to some other related questions.  This includes recent joint work with Martin Fluch.

In this talk I'll explain how to build CAT(0) cube complexes and construct Lipschitz maps between them. The existence of suitable Lipschitz maps is used to prove that the asymptotic dimension of a

CAT(0) cube complex is no more than its dimension.

We analyze the question of the minimal index of a normal subgroup in a free group which does not contain a given element. Recent work by BouRabee-McReynolds and Rivin give estimates for the index. By using results on the length of shortest identities in finite simple groups we recover and improve polynomial upper and lower bounds for the order of the quotient. The bounds can be improved further if we assume that the element lies in the lower central series.