相关论文: A group with deep pockets for all finite generatin…
We prove that any finitely generated one ended group has linear end depth. Moreover, we give alternative proofs to theorems relating the growth of a finitely generated group to the number of its ends.
It is known, that the existence of dead ends (of arbitrary depth) in the Cayley graph of a group depends on the chosen set of generators. Nevertheless there exist many groups, which do not have dead ends of arbitrary depth with respect to…
We show that any group with arbitrarily large finite quotients admits generating sets with respect to which it has arbitrarily large finite dead-end depth. This extends a joint result with Riley and partially answers a question asked there.
The dead-end depth of an element g of a group G, with respect to a generating set A is the distance from g to the complement of the radius $d_A(1,g)$ closed ball, in the word metric $d_A$ defined with respect to A. We exhibit a finitely…
The dead-end depth of an element g of a group with finite generating set A is the distance from g to the complement of the radius d(1,g) closed ball, in the word metric d associated to A. We exhibit a finitely presented group K with two…
Our main result is to show that every infinite, countable, residually finite group $G$ admits a Hausdorff group topology which is neither discrete nor precompact.
For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $A\wr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every…
The complete growth series of a finitely generated group is given by $\sum_{n\ge 0} A_ns^n$, where $A_n$ is the sum of elements of length $n$ in the group semiring. We study the $\mathbb NG$-rationality and $\mathbb NG$-algebraicity of such…
We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that `strongly converge' to the regular representation of the group. The corresponding statement…
We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups.…
The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. We show that every non-positive integer is the…
In SGA 2, Grothendieck conjectures that the \'etale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In…
We study the action of a nilpotent group G with finite generating set S on its horofunction boundary. We show that there is one finite orbit associated to each facet of the polytope obtained by projecting S into the infinite component of…
The higher divergence of a metric space describes its isoperimetric behaviour at infinity. It is closely related to the higher-dimensional Dehn functions, but has more requirements to the fillings. We prove that these additional…
We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to…
We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional…
We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first…
We prove that compact complex manifolds bearing a holomorphic Riemannian metric have infinite fundamental group.
It is proved that any infinite Abelian group of infinite exponent admits a non-discrete reflexive group topology.
We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…