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A nilpotent group without local functional equations for pro-isomorphic subgroups
The pro-isomorphic zeta function of a torsion-free finitely generated
nilpotent group G enumerates finite index subgroups H such that H and G have
isomorphic profinite completions. It admits an Euler product decomposition,
indexed by the rational primes. We manufacture the first example of a
torsion-free finitely generated nilpotent group G such that the local Euler
factors of its pro-isomorphic zeta function do not satisfy functional
equations. The group G has nilpotency class 4 and Hirsch length 25. It is
obtained, via the Malcev correspondence, from a Z-Lie lattice L with a suitable
algebraic automorphism group Aut(L).Comment: 16 page
Detached from their homeland: the Latter-day Saints of Chihuahua, Mexico
Over the past few decades, the homeland concept has received an ever-increasing amount of attention by cultural geographers. While the debate surrounding the necessity and applicability of the concept continues, it is more than apparent that no other geographic term (including culture areas or culture regions) captures the essence of peoples’ attachment to place better than homeland. The literature, however, provides few examples of the deep-seated loyalty people have for a homeland despite being physically detached from that space. Employing land use mapping and informal interviews, this paper seeks to help fill that gap by exemplifying how the daily lives of Mormons living in Chihuahua, Mexico reflect their connection to the United States and the Mormon Homeland. Our research revealed that, among other things, the Anglo residents perpetuate their cultural identity through their unique self-reference, exhibit territoriality links reflected in their built environment, and demonstrate unconditional bonding to their homeland through certain holiday celebrations. It is clear to us, as the Anglo-Mormon experience illustrates, that the homeland concept deserves a place within the geographic lexicon
On pro-isomorphic zeta functions of -groups of even Hirsch length
Pro-isomorphic zeta functions of finitely generated nilpotent groups form one
of the group-theoretic generalisations of the Riemann zeta functions. They are
Dirichlet generating functions enumerating finite-index subgroups whose
profinite completion is isomorphic to that of the ambient group. We study
pro-isomorphic zeta functions of -groups; these form the building blocks
of finitely generated class two nilpotent groups with centre of rank two, up to
commensurability. These groups were classified by Grunewald and Segal, and can
be indexed by primary polynomials whose companion matrices define commutator
relations. We provide a key step towards the elucidation of the pro-isomorphic
zeta functions of -groups of even Hirsch length by describing the
automorphism groups of the associated graded Lie rings. Utilizing this
description of the automorphism groups, we calculate the local pro-isomorphic
zeta functions of groups associated to the polynomials and . In both
cases, the local zeta functions are uniform in the prime~ and satisfy
functional equations.Comment: 29 page
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