Exactness of one relator groups

Abstract

Abstract. A discrete group Γ is C * -exact if the reduced crossed product with Γ converts a short exact sequence of Γ-C * -algebras into a short exact sequence of C * -algebras. A one relator group is a discrete group Γ admitting a presentation Γ = X | R where X is a countable set and R is a single word over X. In this short paper we prove that all one relator discrete groups are C * -exact. Using the Bass-Serre theory we also prove that a countable discrete group Γ acting without inversion on a tree is C * -exact if the vertex stabilizers of the action are C * -exact.

C*-Algebras of Right LCM One-Relator Monoids and Artin–Tits Monoids of Finite Type

“…Consider the case I 2 (m) with m ≥ 2 even. Then (28) gives thatj 0 is the zero map, and consequently K 1 (I ) ∼ = Z and K 0 (I ) ∼ = Z 2 . Moreover, (22) now gives the exact sequence 0 −→ K 1 (C * r (I 2 (m))) −→ K 0 (I ) ι 0 −→ K 0 (A) −→ K 0 (C * λ (I 2 (m))) −→ K 1 (I ) −→ 0.…”

“…This, however, is impossible. Hence P\ {e} = n i=1 a i P. Note that exactness of the one-relator group | u = v follows from [28].…”

“…Recall that˜ = \ {∞} is G-invariant, so that we obtain a partial action G C 0 (˜ ) ⊗ A by restriction. As G is exact [28], we obtain an exact sequence…”

“…if m is odd. Since we have many columns with only one non-zero entry given by 1, we can first perform elementary column operations and then permutations so thatj i is transformed to a matrix whose upper left corner consists of the identity matrix and the lower right corner is of the form ⎛ Adding the third row, multiplied by 1 + β from the left, to the second row yields 1j i withj i as in (28) or (29). All in all, we have transformedj i by performing elementary column and row operations (which correspond to pre-and post-composition with isomorphisms) toj i :…”

Complexes and exactness of certain Artin groups

“…We note that Altobelli characterised the Artin groups of type FC as the smallest class of Artin groups containing the Artin groups of finite type which is closed under amalgamations along parabolic subgroups [1]. Thus, this theorem could alternately be obtained by appealing to the stability theorem for graph products of exact groups first established in [9]. (See also [8] for a more modern discussion.)…”