<p dir="ltr">A group is said to be matricially (Frobenius) stable if every function from the group to unitary matrices that is "almost multiplicative" in the point operator (Frobenius) norm topology is "close" to a genuine unitary representation in the same topology. A result of Dadarlat shows that for a large class of groups, non-torsion even cohomology obstructs matricial stability. However, the proof doesn't generate explicit almost multiplicative maps that are far from genuine representations. In this paper, we compute explicit almost homomorphisms for all finitely generated groups with a non-torsion 2-cohomology class with a residually finite central extension. We use similar techniques to show that finitely generated nilpotent groups are Frobenius stable if and only if they are virtually cyclic, and that a finitely generated group with a non-torsion 2-cohomology class that can be written as a cup product of two 1-cohomology classes is not Frobenius stable.</p><p><br></p>

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Extending an explicit result from Bridson–Conder–Reid, this work provides an algorithm for distinguishing finite quotients between cocompact triangle groups Δ ?and lattices Γ of constant curvature symmetric 2-spaces. Much of our attention will be on when these lattices are Fuchsian groups. We prove that it will suffice to take a finite quotient that is Abelian, dihedral, a subgroup of PSL(<i>n</i>,<b>F</b>_<i>q</i>) (for an odd prime power q), or an Abelian extension of one of these 3 groups. For the latter case, we will require and develop an approach for creating group extensions upon a shared finite quotient of Δ? and Γ which between them have differing degrees of smoothness. Furthermore, on the order of a finite quotient that distinguishes between ?Δ and Γ, we are able to establish an effective upperbound that is superexponential depending on the cone orders appearing in each group.<br>

<p>Whether the growth series of a group is a rational function is investigated in this paper.Parry showed certain torus bundle groups of even trace exhibits rational growth, and thisresult has been extended by the author, Turbo Ho and Mark Pengitore. In this paper, bothresults are combined into a single proof used in [1], and the result is pushed further into thenegative case not covered in earlier works</p>

<p dir="ltr">In this article, we establish results concerning the nilpotent representation theory of groups. In particular, we utilize a theorem of Stallings to provide a general method that constructs pairs of groups that have isomorphic universal nilpotent quotients. We then prove by counterexample that absolute Galois groups of number fields are not determined by their universal nilpotent quotients. We also show that this is the case for residually nilpotent Kleinian groups and in fact, there exist non-isomorphic pairs that have arbitrarily large nilpotent genus. We additionally provide examples of non-isomorphic curves whose geometric fundamental groups have isomorphic universal nilpotent quotients and the isomorphisms are compatible with the outer Galois actions. </p>