Related papers: The Farrell-Jones Conjecture for hyperbolic-by-cyc…

In this article, we give an elementary combinatorial proof of a conjecture about the determination of automorphism group of the power graph of finite cyclic groups, proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.

We prove that residually finite mapping tori of polynomially growing automorphisms of hyperbolic groups, groups hyperbolic relative to finitely many virtually polycyclic groups, right-angled Artin groups (when the automorphism is…

We use the controlled algebra approach to study the problem that whether the Farrell-Jones conjecture is closed under passage to over-groups of finite indices. Our study shows that this problem is closely related to a general problem in…

We develop the theory of $L^2$-torsion of an automorphism of a group and compute it for every automorphism of a group which is hyperbolic and one-ended relative to a finite collection of virtually polycyclic groups. We also prove a…

Using a theorem of L\"uck-Reich-Rognes-Varisco, we show that the Whitehead group of Thompson's group T is infinitely generated, even when tensored with the rationals. To this end we describe the structure of the centralizers and normalizers…

We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.

We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result…

We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general…

This is a survey of the Kawamata-Morrison cone conjecture on the structure of Calabi-Yau varieties and more generally Calabi-Yau pairs. We discuss the proof of the cone conjecture for algebraic surfaces, with plenty of examples. We show…

We prove that there is an algorithm to determine whether a tuple of elements in a toral relatively hyperbolic group G is in the automorphic orbit of the other tuple.

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are…

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively…

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit…

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the…

The descent method is one of the approaches to study the Brauer--Manin obstruction to the local--global principle and to weak approximation on varieties over number fields, by reducing the problem to ``descent varieties''. In recent lecture…

In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chlo\'e Perin that if $H$ is an…

Assuming the classical Farrell-Jones conjecture we produce an explicit (commutative) group ring $R$ and a thick subcategory $\mathsf{C}$ of perfect $R$-complexes such that the Waldhausen $K$-theory space $\mathrm{K}(\mathsf{C})$ is…

We give a concise introduction to the Farrell-Jones Conjecture in algebraic $K$-theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it:…

Given a finitely generated subgroup $\Gamma \le \mathrm{Out}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F} = F_r$, there is a corresponding free group extension $1 \to \mathbb{F} \to E_{\Gamma} \to…

In this paper we show that the fibered isomorphism conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of complex manifolds. A consequence of this…