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The aspherical Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups
The Cavicchioli–Hegenbarth–Repovš generalized Fibonacci groups are defined by the presentations Gn (m, k) = 〈x 1, … , xn | xixi+m = xi+k (1 ⩽ i ⩽ n)〉. These cyclically presented groups generalize Conway's Fibonacci groups and the Sieradski groups. Building on a theorem of Bardakov and Vesnin we classify the aspherical presentations Gn (m, k). We determine when Gn (m, k) has infinite abelianization and provide sufficient conditions for Gn (m, k) to be perfect. We conjecture that these are also necessary conditions. Combined with our asphericity theorem, a proof of this conjecture would imply a classification of the finite Cavicchioli–Hegenbarth–Repovš groups
Finite and infinite quotients of discrete and indiscrete groups
These notes are devoted to lattices in products of trees and related topics.
They provide an introduction to the construction, by M. Burger and S. Mozes, of
examples of such lattices that are simple as abstract groups. Two features of
that construction are emphasized: the relevance of non-discrete locally compact
groups, and the two-step strategy in the proof of simplicity, addressing
separately, and with completely different methods, the existence of finite and
infinite quotients. A brief history of the quest for finitely generated and
finitely presented infinite simple groups is also sketched. A comparison with
Margulis' proof of Kneser's simplicity conjecture is discussed, and the
relevance of the Classification of the Finite Simple Groups is pointed out. A
final chapter is devoted to finite and infinite quotients of hyperbolic groups
and their relation to the asymptotic properties of the finite simple groups.
Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups
updated; more examples added in Section 4; new Proposition 5.1
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