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Register a new userThe conjugacy problem in $GL(n,Z)$
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We present a new algorithm that, given two matrices in $GL(n,Q)$, decides if they are conjugate in $GL(n,Z)$ and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in $GL(n,Z)$ of a matrix in $GL(n,Q)$. We do this by reducing these problems respectively to the isomorphism and automorphism group problems for certain modules over rings of the form $mathcal O_K[y]/(y^l)$, where $mathcal O_K$ is the maximal order of an algebraic number field and $l in N$, and then provide algorithms to solve the latter. The algorithms are practical and our implementations are publicly available in Magma.
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We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves E. Rips problem formulated in 1992. v2: misprints corrected. v3: lemmas 4.7, 4.10 corrected, more misprints fixed.
$Out(F_n):=Aut(F_n)/Inn(F_n)$ denotes the outer automorphism group of the rank $n$ free group $F_n$. An element $phi$ of $Out(F_n)$ is polynomially growing if the word lengths of conjugacy classes in $F_n$ grow at most polynomially under iteration by $phi$. We restrict attention to the subset $UPG(F_n)$ of $Out(F_n)$ consisting of polynomially growing elements whose action on $H_1(F_n, Z)$ is unipotent. In particular, if $phi$ is polynomially growing and acts trivially on $H_1(F_n,Z_3)$ then $phi$ is in $UPG(F_n)$ and also every polynomially growing element of $Out(F_n)$ has a positive power that is in $UPG(F_n)$. In this paper we solve the conjugacy problem for $UPG(F_n)$. Specifically we construct an algorithm that takes as input $phi, psiin UPG(F_n)$ and outputs YES or NO depending on whether or not there is $thetain Out(F_n)$ such that $psi=thetaphitheta^{-1}$. Further, if YES then such a $theta$ is produced.
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes studied independently by Haglund and Wise.
In this paper, we consider the conjugacy growth function of a group, which counts the number of conjugacy classes which intersect a ball of radius $n$ centered at the identity. We prove that in the case of virtually polycyclic groups, this function is either exponential or polynomially bounded, and is polynomially bounded exactly when the group is virtually nilpotent. The proof is fairly short, and makes use of the fact that any polycyclic group has a subgroup of finite index which can be embedded as a lattice in a Lie group, as well as exponential radical of Lie groups and Dirichlets approximation theorem.
In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.
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