Solving decision problems in finitely presented groups via generalized small cancellation theory

Abstract

This thesis studies two decision problems for finitely presented groups. Using a standard RAM model of computation, in which the basic arithmetical operations on integers are assumed to take constant time, in Part I we develop an algorithm IsConjugate, which on input a (finite) presentation defining a hyperbolic group 𝐺, correctly decides whether 𝑤₁ ϵ 𝑋* and 𝑤₂ ϵ 𝑋* are conjugate in 𝐺, and if so, then for each 𝑖 ϵ {1,2}, returns a cyclically reduced word 𝑟ᵢ that is conjugate in 𝐺 to 𝑤ᵢ, and an 𝑥 ϵ 𝑋* such that r₂= G 𝑥^{-1} r_1 x (hence if 𝑤₁ and 𝑤₂ are already cyclically reduced, then it returns an 𝑥 ϵ 𝑋* such that 𝑤₂=_G x^{-1} w_1 x). Moreover, IsConjugate can be constructed in polynomial-time in the input presentation < 𝑋 | 𝑅 >, and IsConjugate runs in time O((|𝑤₁| + |𝑤₂| min{|𝑤₁|, |𝑤₂|}). IsConjugate has been implemented in the MAGMA software, and our experiments show that the run times agree with the worst-case time complexities. Thus, IsConjugate is the most efficient general practically implementable conjugacy problem solver for hyperbolic groups. It is undecidable in general whether a given finitely presented group is hyperbolic. In Part II of this thesis, we present a polynomial-time procedure VerifyHypVertex which on input a finite presentation for a group G, returns true only if G is hyperbolic. VerifyHypVertex generalizes the methods from [34], and in particular succeeds on all presentations on which the implementation from [34] succeeds, and many additional presentations as well. The algorithms have been implemented in MAGMA, and the experiments show that they return a positive answer on many examples on which other comparable publicly available methods fail, such as KBMAG.