Initially, we set out to construct non-uniform ‘arithmetic’ lattices in Kac-Moody groups of rank 2 over finite fields, as constructed by Tits ([Ti1], [Ti2]) using the BruhatTits tree of a Tits system for such groups. This attempt succeeded, and in fact, the construction we used can be applied to higher rank Kac-Moody groups over sufficiently large finite fields, and their buildings (Theorem 1.7...

Abstract. Hausdorff’s gap condition was satisfied by his original 1936 construction of an (ω1, ω1) gap in P(N) / Fin. We solve an open problem in determining whether Hausdorff’s condition is actually stronger than the more modern indestructibility condition, by constructing an indestructible (ω1, ω1) gap not equivalent to any gap satisfying Hausdorff’s condition, from uncountably many random re...

We say that a partial word w over an alphabet A is square-free if every factor xx of w such that x and x are compatible is either of the form ⋄a or a⋄ where ⋄ is a hole and a ∈ A. We prove that there exist uncountably many square-free partial words over a ternary alphabet with an infinite number of holes.

In this paper we prove the existence of uncountably many indecomposable continua in the dynamics of complex exponentials of the form Eλ(z) = λe z with λ > 1/e. These continua contain points that share the same itinerary under iteration of Eλ. These itineraries are bounded but consist of blocks of 0’s whose lengths increase, and hence these continua are never periodic.

We investigate some situation in which automorphisms of a groupG are uniquely determined by their restrictions to a proper subgroup H . Much of the paper is devoted to studying under which additional hypotheses this property forces G to be nilpotent if H is. As an application we prove that certain countably infinite locally nilpotent groups have uncountably many (outer) automorphisms.

Let E be a coanalytic equivalence relation on a Polish space X and (An)n∈ù a sequence of analytic subsets of X . We prove that if lim supn∈K An meets uncountably many E-equivalence classes for every K ∈ [ù] , then there exists a K ∈ [ù] such that T n∈K An contains a perfect set of pairwise E-inequivalent elements. §

A widely accepted method to specify (possibly infinite) behaviour is to define it as the solution, in some process algebra, of a recursive specification, i.e., a system of recursive equations over the fundamental operations of the process algebra. The method only works if the recursive specification has a unique solution in the process algebra; it is well-known that guardedness is a sufficient ...

Abstract We show that Gromov’s monsters arising from i.i.d. random labellings of expanders (that we call monsters) have linear divergence along a subsequence, so in particular they do not contain Morse quasigeodesics, and are quasi-isometric to graphical small cancellation expanders. Moreover, by further studying the function, there uncountably many quasi-isometry classes monsters.