Abstract:
We extend the structure theorem for the subgroups of the class of HNN groups to a new class of groups called quasi-HNN groups. The main technique used is the subgroup theorem for groups acting on trees with inversions.

Abstract:
A group $G$ is said to have the Howson property (or to be a Howson group) if the intersection of any two finitely generated subgroups of $G$ is finitely generated subgroup. It is proved that descending HNN-extension is not a Howson group under some assumptions satisfied by the base group of HNN-extension. In particular, a result of the paper joined with a Burns - Brunner result (received in 1979) implies that any descending HNN-extension of non-cyclic free group does not have the Howson property.

Abstract:
An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.

Abstract:
As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures. This new point of view also allows us to extend, and simplify the proofs of, some cases where these conjectures were known to be true.

Abstract:
We show that Serre's Intersection Multiplicity Conjecture holds for a formal power series ring A over a complete, two-dimensional regular local ring R. From this, we deduce the corresponding result for the local rings of any scheme X which is a smooth extension of a regular, two-dimensional base Y. We also investigate the connection between intersection multiplicity and transversality in the unramified setting via a local analysis on the blowup.

Abstract:
Reduced HNN extensions of von Neumann algebras (as well as $C^*$-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.

Abstract:
We present sufficient conditions for HNN extensions to be inner amenable, respectively ICC, which give necessary and sufficient criteria among Baumslag-Solitar groups. We deduce that such a group, viewed as acting on its Bass-Serre tree, contains non trivial elements which fix unbounded subtrees.

Abstract:
Using recent developments on locally compact groups, we are able to obtain quantitative results on embeddings into Lebesgue spaces for a large class of HNN extensions.

Abstract:
Galois conjugation relates unitary conformal field theories (CFTs) and topological quantum field theories (TQFTs) to their non-unitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois conjugated Hamiltonians are typically non-Hermitian, we find that their ground state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such non-unitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this we rigorously prove that no local change of basis can transform the ground states of the Galois conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasi-local Hamiltonians. A similar statement holds for many other non-unitary TQFTs. One consequence is that the "Gaffnian" wave function cannot be the ground state of a gapped fractional quantum Hall state.