arXiv research

Proves A-theoretic Farrell-Jones conjecture for virtually solvable groups.

New representation shows non virtually solvable subgroups of mapping class groups have non virtually solvable elements.

We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for virtually solvable groups.

New proof shows RFRS groups virtually fibred if their $L^2$-Betti number vanishes.

Study proves surface group quotients have non-virtually solvable profinite completions.

Let $p$ be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually $p$. Let $M=M^3$ be a virtually fibered 3-manifold. It is well-known that $G=π_1(M)$ is residually solvable and even residually finite solvable. We prove that $G$ is always virtually residually $p$…

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties o…

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic…

In this paper, we prove the Farrell-Jones Conjecture for the solvable Baumslag-Solitar groups with coefficients in an additive category. We also extend our results to groups of the form, Z[1/p] semidirect product with any virtually cyclic group, where p is a prime number.

We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finite…

Unified algebraic framework for virtual braid structures with strong structural consequences.

Extends confining subset theory to describe hyperbolic actions of solvable groups with higher rank abelianizations.

A classification of partially hyperbolic diffeomorphisms on 3-dimensional manifolds with (virtually) solvable fundamental group is obtained. If such a diffeomorphism does not admit a periodic attracting or repelling two-dimensional torus, it is dynamically coherent and leaf conjugate to a known algebraic example. This …

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical …

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may dec…

We show that the group cohomology of torsion-free virtually polycyclic groups and the continuous cohomology of simply connected solvable Lie groups can be computed by the rational cohomology of algebraic groups. Our results are generalizations of certian results on the cohomology of solvmanifolds and infra-solvmanifold…

The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example for discrete subgroups of Lie groups, virtually poly-infinite cyclic groups, Artin braid groups, a class of virtually poly-surface groups and virtually solvable lin…

We describe the BNS invariant of Kaehler groups. As an application we prove that if the fundamental group of a Kaehler manifold is solvable, it is virtually nilpotent.

Through highly non-constructive methods, works by Bestvina, Culler, Feighn, Morgan, Paulin, Rips, Shalen, and Thurston show that if a finitely presented group does not split over a virtually solvable subgroup, then the space of its discrete and faithful actions on hyperbolic n-space, modulo conjugation, is compact for …

A companion result of the the Tits alternative for $Out(F_n)$ is proved: Every solvable subgroup of $Out(F_n)$ is finitely generated and virtually abelian.

Study geometric actions of groups on horocyclic products.

Let $G$ be a simply connected, solvable Lie group and $Γ$ a lattice in $G$. The deformation space $\mathcal{D}(Γ,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(Γ,G)$ of all lattice embeddings of $Γ$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev…

Characterizes holonomies of convex projective cusps.

We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we show that the algebraic entropy of discrete subgroups of a given Lie group is unif…

We show that for a Lie group $G=\R^{n}\ltimes_φ \R^{m}$ with a semisimple action $φ$ which has a cocompact discrete subgroup $Γ$, the solvmanifold $G/Γ$ admits a canonical invariant formal (i.e. all products of harmonic forms are again harmonic) metric. We show that a compact oriented aspherical manifold of dimension l…

New findings on mapping class group actions on the circle, improving critical regularity.

The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.

Proves conditions for nearby special Lagrangians in Calabi-Yau manifolds.

The study proves theorems about quasiflats in hierarchically hyperbolic spaces.

In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ wh…

Abstract: Study of 2-knot groups with restrictions on normal subgroups.

Non-coherence proven for certain groups with specific mapping tori.

Finite subgroups of good groups correspond to their profinite completions.

A Garside group is a group admitting a finite lattice generating set D. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π,1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice D, and there is a simple suff…

The article has been withdrawn by the author. Wolfgang Lueck and Peter Linnell pointed out that the proof of Lemma 3.8 does not apply to the unrestricted case of wreath product. It is not clear at this stage how to complete the proof of Theorem 3.1 using the present version of Lemma 3.8. The valid results originating f…

Study on solvable Lie groups with specific Weyl connections.

Extended logarithm for solvable elements in mapping class groups.

For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$-trees, using stretching factors of $G$-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in t…

Proves non-solvability of concordance groups using Milnor invariants.

Groups of 2-stratifolds have a solvable word problem.

Classifies Ricci soliton subgroups in specific Lie groups.

We prove that all elements of infinite order in $Out(F_n)$ have positive translation lengths; moreover, they are bounded away from zero. Consequences include a new proof that solvable subgroups of $Out(F_n)$ are finitely generated and virtually abelian and the new result that such subgroups are quasi-convex.

It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.

New virtual version of Thompson's group created to handle virtual knots.

Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.

The paper characterizes crystallographic groups derived from virtual braid and twin groups.

Counterexample found for Stein property of certain solvable Lie groups.

Study virtual braid groups, proving a key subgroup result.