Non-triviality of the Poisson boundary of random walks on the group of Monod

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$ , we prove that either $H$ is solvable or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group $F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].

DIRECT SUMS OF H-SUPPLEMENTED MODULES

A module M is said to be H-supplemented if, for any submodule X of M, there exists a direct summand M′ of M such that M = X + Y if and only if M = M′ + Y for all Y ⊆ M (cf. [S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Mathematical Society Lecture Note Series, Vol. 147 (Cambridge University Press, 1999)]). We say that a module M is semi-lifting if any direct summand of M is H-supplemented. A H-supplemented module is a dual to a Goldie-extending module which was introduced by Akalan–Birkenmeier–Tercan [Goldie extending modules, Comm. Algebra37 (2009) 663–683]. In this paper, we give some characterizations of semi-lifting modules and H-supplemented modules. In addition, we consider generalizations of relative projectivities and apply them to the study of direct sums of semi-lifting modules.

ON STOCHASTIC DOMINANCE OF NILPOTENT OPERATORS

In this paper, motivated by Nica and Speicher [Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, Vol. 335 (Cambridge University Press, 2006)] and Kubo and Kuo [MRM-factors for the probability measures in the Meixner class, Infin. Dimens. Anal. Quantum Probab. Relat. Top.13 (2010) 525–550], we characterize a particular nilpotent case of a truncated forward shift operator by applying the technique of the random walks with repeated reflections and associated renewal equations. We also establish a stochastic order relationship by applying the crossing criteria.

ON THE EXISTENCE OF NON-ABELIAN (210, 77, 28), (336, 135, 54) AND (496, 55, 6) DIFFERENCE SETS

Lander [Symmetric Design: An Algebraic Approach, London Mathematical Society Lecture Note Series, Vol. 74 (Cambridge University Press, Cambridge, 1983)] showed that some abelian (v, k, λ) difference sets do not exist in some groups of order v and listed some parameters of difference sets that were open for small values of k (k ≤ 50). Various authors have since studied the open cases and have either constructed the difference sets when they exist or proved their non-existence. Using restrictions imposed by the underlying normal subgroups of groups, algebraic number theory, group representations and rational idempotents of the group ring, we rule out the existence of (496, 55, 6) difference sets and show that non-abelian (210, 77, 28) and (336, 135, 54) difference sets do not exist in most groups of 210 and 336, respectively.