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29
Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems
, 2004
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Tree Actions of Automorphism Groups
 J. Group Theory
"... We introduce conditions on a group action on a tree that are sufficient for the action to extend to the automorphism group. We apply this to two different classes of onerelator groups: certain BaumslagSolitar groups and onerelator groups with centre. In each case we derive results about the autom ..."
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Cited by 15 (0 self)
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We introduce conditions on a group action on a tree that are sufficient for the action to extend to the automorphism group. We apply this to two different classes of onerelator groups: certain BaumslagSolitar groups and onerelator groups with centre. In each case we derive results about the automorphism group, and deduce that there are relatively few outer automorphisms.
On the automorphism group of generalized BaumslagSolitar groups
"... Abstract. A generalized BaumslagSolitar 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 nonabelian free groups or is virtually nilpotent of class ≤ 2. It has torsion only at finitely ma ..."
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Abstract. A generalized BaumslagSolitar 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 nonabelian free groups or is virtually nilpotent of class ≤ 2. It has torsion only at finitely many primes. One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent. If G is unimodular (virtually Fn × Z), then Out(G) is commensurable with a semidirect product Z k ⋊Out(H) with H virtually free. Contents
Some connections between residual finiteness, finite embeddability and the word problem
 J. London Math. Soc
, 1969
"... We prove in this note that, in a variety V, residual finiteness of a finitely presented algebra A is equivalent to the property that any finite partial algebra contained in A is embeddable in a finite Falgebra and each implies that A has a solvable word problem. Finite embeddability. An algebra A i ..."
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Cited by 9 (0 self)
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We prove in this note that, in a variety V, residual finiteness of a finitely presented algebra A is equivalent to the property that any finite partial algebra contained in A is embeddable in a finite Falgebra and each implies that A has a solvable word problem. Finite embeddability. An algebra A is residually finite if for any x # y in A, there is a homomorphism a of A onto a finite algebra such that xct # yu. For the notion of an incomplete or partial algebra in a variety, we refer to [4, 6]. We say that an algebra A in a variety V has the finite embeddability property if any finite incomplete Falgebra contained in A is embeddable in a finite Falgebra. A variety V is said to have the finite embeddability property if every algebra in V has the property. Thus, a variety V has the finite embeddability property if any finite incomplete Kalgebra which is embeddable is embeddable in a finite Falgebra. We note also that a variety has the finite embeddability property if its finitely generated algebras have this property. To see this, let A be an algebra in a variety V whose finitely generated algebras have the finite embeddability property and let / be a finite incomplete algebra
Residual amenability and the approximation of L 2 –invariants
 Michigan Math. J
, 1999
"... Abstract. We generalize Lück’s Theorem to show that the L 2Betti numbers of a residually amenable covering space are the limit of the L 2Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant cla ..."
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Cited by 6 (0 self)
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Abstract. We generalize Lück’s Theorem to show that the L 2Betti numbers of a residually amenable covering space are the limit of the L 2Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant class, and that the L 2 torsion is a homotopy invariant for such spaces. We give examples of residually amenable groups, including the BaumslagSolitar groups.
Asymptotic cyclic expansion and bridge groups of formal proofs
 JOURNAL OF ALGEBRA
, 2001
"... Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify th ..."
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Cited by 5 (1 self)
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Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs.
Splittings of generalized Baumslag–Solitar groups
, 2006
"... Abstract. We study the structure of generalized Baumslag–Solitar groups from the point of view of their (usually nonunique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (fully reduced decompositions) and ..."
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Cited by 5 (1 self)
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Abstract. We study the structure of generalized Baumslag–Solitar groups from the point of view of their (usually nonunique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (fully reduced decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag–Solitar groups with no nontrivial integral moduli.
Deformation Spaces of G–trees and Automorphisms of Baumslag–Solitar groups
, 2008
"... We construct an invariant subcomplex of a deformation space of G– trees. We show that this subcomplex is finite dimensional in certain cases and provide an example that is not finite dimensional. Using this subcomplex we compute the automorphism group of the classical nonsolvable Baumslag–Solitar gr ..."
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We construct an invariant subcomplex of a deformation space of G– trees. We show that this subcomplex is finite dimensional in certain cases and provide an example that is not finite dimensional. Using this subcomplex we compute the automorphism group of the classical nonsolvable Baumslag–Solitar groups BS(p, q). The most interesting case is when p properly divides q. Collins and Levin computed a presentation for Aut(BS(p, q)) in this case using algebraic methods. Our computation uses Bass–Serre theory to derive these presentations. Additionally, we provide a geometric argument showing Out(BS(p, q)) (and hence Aut(BS(p, q))) is not finitely generated when p properly divides q. Baumslag–Solitar groups have the following standard presentations: BS(p, q) = 〈x, t  tx p t −1 = x q 〉. (1) We will see that when p properly divides q there are infinitely many similar presentations showing some hidden structure to this groups. These groups were
Counting homomorphisms onto finite solvable groups
 Journal of Algebra
, 2005
"... We present a method for computing the number of epimorphisms from a finitelypresented group G to a finite solvable group \Gamma, which generalizes a formula of Gasch\"utz. Key to this approach are the degree 1 and 2 cohomology groups of G, with certain twisted coefficients. As an application, we co ..."
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We present a method for computing the number of epimorphisms from a finitelypresented group G to a finite solvable group \Gamma, which generalizes a formula of Gasch\"utz. Key to this approach are the degree 1 and 2 cohomology groups of G, with certain twisted coefficients. As an application, we count lowindex subgroups of G. We also investigate the finite solvable quotients of the BaumslagSolitar groups, the Baumslag parafree groups, and the Artin braid groups.