Betti numbers of finitely presented groups and very rapidly growing functions

Define the length of a finite presentation of a group G as the sum of lengths of all relators plus the number of generators. How large can the kth Betti number b_k(G)= rank H_k(G) be providing that G has length ≤N and b_k(G) is finite? We prove that for every k≥3 the maximum b_k(N) of the kth Betti numbers of all such groups is an extremely rapidly growing function of N. It grows faster that all functions previously encountered in mathematics (outside of logic) including non-computable functions (at least those that are known to us). More formally, b_k grows as the third busy beaver function that measures the maximal productivity of Turing machines with ≤N states that use the oracle for the halting problem of Turing machines using the oracle for the halting problem of usual Turing machines.We also describe the fastest possible growth of a sequence of finite Betti numbers of a finitely presented group. In particular, it cannot grow as fast as the third busy beaver function but can grow faster than the second busy beaver function that measures the maximal productivity of Turing machines using an oracle for the halting problem for usual Turing machines. We describe a natural problem about Betti numbers of finitely presented groups such that its answer is expressed by a function that grows as the fifth busy beaver function.Also, we outline a construction of a finitely presented group all of whose homology groups are either or trivial such that its Betti numbers form a random binary sequence.     

Asymptotic dimension of finitely presented groups

We prove that if a finitely presented group is one-ended, then its asymptotic dimension is greater than . It follows that a finitely presented group of asymptotic dimension is virtually free.

     

Seperating the intrinsic complexity and the derivational complexity of the word problem for finitely presented groups

A pseudo-natural algorithm for the word problem of a finitely presented group is an algorithm which not only tells us whether or not a word w equals 1 in the group but also gives a derivation of 1 from w when w equals 1. In [13], [14] Madlener and Otto show that, if we measure complexity of a primitive recursive algorithm by its level in the Grzegorczyk hierarchy, there are groups in which a pseudo-natural algorithm is arbitrarily more complicated than an algorithm which simply solves the word problem. In a given group the lowest degree of complexity that can be realised by a pseudo-natural algorithm is essentially the derivational complexity of that group. Thus the result separates the derivational complexity of the word problem of a finitely presented group from its intrinsic complexity. The proof given in [13] involves the construction of a finitely presented group G from a Turing machine T such that the intrinsic complexity of the word problem for G reflects the complexity of the halting problem of T, while the derivational complexity of the word problem for G reflects the runtime complexity of T. The proof of one of the crucial lemmas in [13] is only sketched, and part of the purpose of this paper is to give the full details of this proof. We will also obtain a variant of their proof, using modular machines rather than Turing machines. As for several other results, this simplifies the proofs considerably. MSC: 03D40, 20F10.     

Subgroups Of Direct Products Of Elementarily Free Groups

The structure of groups having the same elementary theory as free groups is now known: they and their finitely generated subgroups form a prescribed subclass of the hyperbolic limit groups. We prove that if G ₁,...,G _n are in then a subgroup Γ ⊂ G ₁ × … × G _n is of type FP _n if and only if Γ is itself, up to finite index, the direct product of at most n groups from . This provides a partial answer to a question of Sela. This work was supported in part by Franco–British Alliance project PN 05.004. The first author is also supported by an EPSRC Senior Fellowship and a Royal Society Wolfson Research Merit Award. Received: July 2005 Accepted: April 2006     

Double Coset Decompositions of Groups

We show that residually finite or word hyperbolic groups which can be decomposed as a finite union of double cosets of a cyclic subgroup are necessarily virtually cyclic, and we apply this result to the study of Frobenius permutation groups. We show that in general, finite double coset decompositions of a group can be interpreted as an obstruction to splitting a group as a free product with amalgamation or an HNN extension.     

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that and let Γ be a uniform lattice in G.

(a)

If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.

(b)

If CH fails, then Γ has 2^2ω asymptotic cones up to homeomorphism.

     

On linearity of finitely generated R-analytic groups

In this paper we prove that if R is a commutative Noetherian local pro-p domain of characteristic 0, then every finitely generated R-standard group is linear. This work has been partially supported by the FEDER, the MEC Grant MTM2004-04665 and the Ramón y Cajal Program.     

Subdirect products of finitely presented metabelian groups

There has been substantial investigation in recent years of subdirect products of limit groups and their finite presentability and homological finiteness properties. To contrast the results obtained for limit groups, Baumslag, Bridson, Holt and Miller investigated subdirect products (fibre products) of finitely presented metabelian groups. They showed that, in contrast to the case for limit groups, such subdirect products could have diverse behaviour with respect to finite presentability.We show that, in a sense that can be made precise, ‘most’ subdirect products of a finite set of finitely presented metabelian groups are again finitely presented. To be a little more precise, we assign to each subdirect product a point of an algebraic variety and show that, in most cases, those points which correspond to non-finitely presented subdirect products form a subvariety of smaller dimension.     

The periodic groups saturated by finitely many finite simple groups

Denote by $\mathfrak{M}$ the set whose elements are the simple 3-dimensional unitary groups U ₃(q) and the linear groups L ₃(q) over finite fields. We prove that every periodic group, saturated by the groups of a finite subset of $\mathfrak{M}$ , is finite.     

Preservation and distortion of area in finitely presented groups

IfK=G where is a tame automorphism of the 1-relator groupG, then the combinatorial area of loops in a Cayley graph ofG is undistorted in a Cayley graph ofK. Examples of distortion of area in fibres of fibrations over the circle are given and a notion of exponent of area distortion is introduced and studied. The inclusion of a finitely generated abelian subgroup in the fundamental group of a compact 3-manifold does not distort area.Partially supported by NSF grant DMS-9200433.     

A Geometric Proof of Stallings' Theorem on Groups with More than One End

Stallings showed that a finitely generated group which has more than one end splits as an amalgamated free product or an HNN extension over a finite subgroup. Dunwoody gave a new geometric proof of the theorem for the class of almost finitely presented groups, and separately, using somewhat different methods, generalised it to a larger class of splittings. Here we adapt the geometric method to the class of finitely generated groups using Sageev's generalisation of Bass Serre theory concerning group pairs with more than one end, and show that this new proof simultaneously establishes Dunwoody's generalisation.     

On monomial representations of finitely generated nilpotent groups

A result of Segal states that every complex irreducible representation of a finitely generated nilpotent group G is monomial if and only if G is abelian-by-finite. A conjecture of Parshin, recently proved affirmatively by Beloshapka and Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture using ideas of Kutzko and Brown. We also give a characterization of the finite-dimensional irreducible representations of two-step nilpotent groups and describe these completely for two-step groups whose center has rank one.