We introduce a natural generalization of Borel’s Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCא0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture,¬BCא1 is equivalent to the existence of a Kurepa tree of heightא1. Using the connection of BCκ with a generalization of Kurepa’s Hypothesis, we obtain the following...

We prove analogues for reductive algebraic groups of some results for finite groups due to Knörr and Robinson from ‘Some remarks on a conjecture of Alperin’, J. London Math. Soc (2) 39 (1989), 48–60, which play a central rôle in their reformulation of Alperin’s conjecture for finite groups.

Given a commutative unital ring $R$, we show that the finiteness length of group $G$ is bounded above by Borel subgroup rank one $\mathbf{B}_2^\circ(R)=\left( \begin{smallmatrix} * & \\ 0 \end{smallmatrix} \right)\leq\mathrm{SL}_2(R)$ whenever admits certain $R$-representations with metabelian image. Combined results due to Bestvina--Eskin--Wortman and Gandini, this gives new proof (a generaliz...

A famous conjecture (usually called Ryser’s conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an r-uniform r-partite hypergraph H, the inequality τ(H) 6 (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem, and in the case of r = 3, which was proved by Aharoni in 2001. Here we ...

The aim of this paper is to prove that, if R is a commutative regular ring in which 2 is a unit, then the reduced theory of quadratic forms with invertible coefficients in R, modulo a proper preorder T , satisfies Marshall’s signature conjecture and Milnor’s Witt ring conjecture (for precise statements, see Section 1 below). For that purpose we use the theory of special groups (abbreviated SG),...

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The algebraicity conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been Borovik’s program of transferring methods from finite group theory. Borovik’s program has led to consi...

in this paper, we investigate the zassenhaus conjecture for $psl(4,3)$ and $psl(5,2)$. consequently, we prove that the prime graph question is true for both groups.