We make a comprehensive and self-contained study of compact bicrossed products arising from matched pairs of discrete groups and compact groups. We exhibit an automatic regularity property of such a matched pair and describe the representation theory and the fusion rules of the associated bicrossed product G. We investigate the relative co-property (T ) and the relative co-Haagerup property of the pair comprising of the compact group and the bicrossed product, discuss property (T ) and Haagerup property of the discrete dual G, and review co-amenability of G as well. We distinguish two such non-trivial compact bicrossed products with relative coproperty (T ) and also provide an infinite family of pairwise non isomorphic non-trivial discrete quantum groups with property (T ), the existence of even one of the latter was unknown. Finally, we examine all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also establish the permanence of rapid decay and weak amenability and provide several explicit examples.

We introduce a class of automorphisms of compact quantum groups (CQGs) which may be thought of as inner automorphisms and explore the behavior of normal subgroups of CQGs under these automorphisms. We also define the notion of center of a CQG and compute the center for several examples. We briefly touch upon the commutator subgroup of a CQG and discuss how its relation with the center can be different from the classical case.

Associated to any compact quantum group G ⊂ U + N is a canonical family of group dual subgroups Γ Q ⊂ G, parametrized by unitaries Q ∈ U N , playing the role of "maximal tori" for G. We present here a series of conjectures, relating the various algebraic and analytic properties of G to those of the family { Γ Q |Q ∈ U N }.2000 Mathematics Subject Classification. 46L65.

This paper initiates the study of non-commutative dynamical systems of the form (G, Γ), where a discrete group Γ acts on a compact quantum group (CQG) G by quantum automorphisms. We obtain combinatorial conditions for such dynamical systems to be ergodic, mixing, compact, etc. and provide a wide variety of examples to illustrate these conditions. We generalize a well-known theorem of Halmos to demonstrate 'reversal of arrows' in the ergodic hierarchy relevant to the context and make a study of spectral measures for actions of (non-commutative) groups. We investigate the structure of such dynamical systems and under certain restrictions exhibit the existence and uniqueness of the maximal ergodic invariant normal subgroup of such systems. As an application, we study the size of normalizing algebras of masas arising from groups in von Neumann algebraic CQGs and show that the normalizing algebra of such masas are the von Neumann algebras generated by co-commutative CQGs.

The bacterial cell wall is primarily composed of a mesh of stiff glycan strands cross-linked by peptide bridges and is essential for safeguarding the cell. The structure of the cell wall has to be stiff enough to bear the high turgor pressure and sufficiently tough to ensure protection against failure.Here we explore the role of various design features of the cell in enhancing the toughness of the cell wall. We explain how the glycan strand length distribution and the degree of cross-linking can play a vital role in ensuring that the cell wall offers sufficient resistance to propagation of cracks. We suggest a possible mechanism by which peptide bond hydrolysis can also help mitigate this risk of failure. We also study the reinforcing effect of MreB on the cell wall and conclude that the cross-linked structure of the cell wall plays the more important role in safeguarding against mechanical failure due to cracking..