A non-LEA Sofic Group

We describe elementary examples of finitely presented sofic groups which are not residually amenable (and thus not initially subamenable or LEA, for short). We ask if an amalgam of two amenable groups over a finite subgroup is residually amenable and answer this positively for some special cases, including countable locally finite groups, residually nilpotent groups and others.Comment: The main theorem is strengthened so that the Sofic examples are shown to have no co-amenable LEA subgroup

Asymptotically CAT(0) groups

We develop a general theory for asymptotically CAT(0) groups; these are groups acting geometrically on a geodesic space, all of whose asymptotic cones are CAT(0)

One Relator Quotients of Graph Products

In this paper, we generalise Magnus' Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups

Ping pong on CAT(0) cube complexes

Let $G$ be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex $X$ without fixed points at infinity. We show that for any finite collection of simultaneously inessential subgroups $\{H_1, \ldots, H_k\}$ in $G$, there exists an element $g$ of infinite order such that $\forall i$, $\langle H_i, g\rangle \cong H_i * \langle g\rangle$. We apply this to show that any group, acting faithfully and geometrically on a non-Euclidean possibly reducible CAT(0) cube complex, has property $P_{naive}$ i.e. given any finite list $\{g_1, \ldots, g_k\}$ of elements from $G$, there exists $g$ of infinite order such that $\forall i$, $\langle g_i, g\rangle \cong \langle g_i \rangle *\langle g\rangle$. This applies in particular to the Burger-Moses simple groups that arise as lattices in products of trees. The arguments utilize the action of the group on its Poisson boundary and moreover, allow us to summarise equivalent conditions for the reduced $C^*$-algebra of the group to be simple

2D problems in groups

We investigate a conjecture about stabilisation of deficiency in finite index subgroups and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem. We verify the pro-$p$ version of the conjecture, as well as its higher dimensional abstract analogues.Comment: comments welcome, some references added in v

Asymptotically CAT(0) groups

We develop a general theory for asymptotically CAT(0) groups; these are groups acting geometrically on a geodesic space, all of whose asymptotic cones are CAT(0)

On Property (FA) for wreath products

We characterize permutational wreath products with Property (FA). For instance, the standard wreath product A wr B of two nontrivial countable groups A,B, has Property (FA) if and only if B has Property (FA) and A is a finitely generated group with finite abelianisation. We also prove an analogous result for hereditary Property (FA). On the other hand, we prove that many wreath products with hereditary Property (FA) are not quotients of finitely presented groups with the same property.Comment: 12 pages, 0 figur

Rayleigh quotients of Dillon's functions

The Walsh--Hadamard spectrum of a bent function uniquely determines a dual function. The dual of a bent function is also bent. A bent function that is equal to its dual is called a self-dual function. The Hamming distance between a bent function and its dual is related to its Rayleigh quotient. Carlet, Danielsen, Parker, and Sole studied Rayleigh quotients of bent functions in ${\mathcal PS}_{ap}$, and obtained an expression in terms of a character sum. We use another approach comprising of Desarguesian spreads to obtain the complete spectrum of Rayleigh quotients of bent functions in $\mathcal{PS}_{ap}$