In 1987, Gromov conjectured that for every non-elementary hyperbolic group G there is an n = n(G) such that the quotient group G/Gn is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of G/Gn is given, it is proven that the word and conjugacy problem are solvable in G/Gn and that ⋂∞ k=1G k = {1}. The proofs heavily depend upon prior authors’ results...

Yasutaka Nakanishi asked in 1981 whether a 3-move is an unknotting operation. In Kirby’s problem list, this question is called The Montesinos-Nakanishi 3-move conjecture. We define the nth Burnside group of a link and use the 3rd Burnside group to answer Nakanishi’s question. One of the oldest elementary formulated problems in classical Knot Theory is the 3move conjecture of Nakanishi. A 3-move...

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve.

Given a periodic quotient of torsion-free hyperbolic group, we provide fine lower estimate the growth function any sub-semi-group. This generalizes results Razborov and Safin for free groups.

Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K stretchy="false">) \mathrm{GL}(n,K) , where