Related papers: One-relator Kaehler groups

We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one…

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$…

We prove that if the fundamental group of an arbitrary three-manifold -- not necessarily closed, nor orientable -- is a Kaehler group, then it is either finite or the fundamental group of a closed orientable surface.

Let $ 1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ be an exact sequence of finitely presented groups where Q is infinite and not virtually cyclic, and is the fundamental group of some closed 3-manifold. If G is Kaehler, we…

We give a criterion of the existence of a presentation with a single relation for the commutator subgroup $RC_{\mathscr{K}}'$ of a right-angled Coxeter group $RC_{\mathscr{K}}$. Namely, we prove that $RC_{\mathscr{K}}'$ is a one-relator…

A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the…

A compact K\"ahler manifold is shown to be simply-connected if its `symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective…

We prove that one-relator groups are coherent, solving a well-known problem of Gilbert Baumslag. Our proof strategy is readily applicable to many classes of groups of cohomological dimension two. We show that fundamental groups of…

We show that any one-relator group $G=F/\langle\langle w\rangle\rangle$ with torsion is coherent -- i.e., that every finitely generated subgroup of $G$ is finitely presented -- answering a 1974 question of Baumslag in this case.

We prove that a K\"ahler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free…

A class of one-relator groups such that every group in the class is determined by a triple of integers and is an HNN-extension of some Baumslag -- Solitar group is considered. A criterion for two groups in this class to be isomorphic and…

A one-relator group is a group $G_r$ that admits a presentation $\langle S \mid r \rangle$ with a single relation $r$. One-relator groups form a rich classically studied class of groups in Geometric Group Theory. If $r \in F(S)'$, the…

For every pair of positive integers $p > q$ we construct a one-relator group $R_{p,q}$ whose Dehn function is $\simeq n^{2 \alpha}$ where $\alpha = \log_2(2p / q)$. The group $R_{p,q}$ has no subgroup isomorphic to a Baumslag-Solitar group…

How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group $G$ is said to be properly 3-realizable if there…

A linear algebraic group G is over a field K is called a Cayley K-group if it admits a Cayley map, i.e., a G-equivariant K-birational isomorphism between the group variety G and its Lie algebra. We classify real reductive algebraic groups…

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain…

We prove that with probability tending to 1, a 1-relator group with at least 3 generators and relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both…

The general {\bf surface group conjecture} asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related…

The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G can be realized as both the fundamental group of a closed 3-manifold and of a compact K\"ahler…

We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed…