# Degrees of rigidity for Souslin trees

• G. Fuchs and J. D. Hamkins, “Degrees of rigidity for Souslin trees,” Journal of Symbolic Logic, vol. 74, iss. 2, pp. 423-454, 2009.
@ARTICLE{FuchsHamkins2009:DegreesOfRigidity,
AUTHOR = {Fuchs, Gunter and Hamkins, Joel David},
TITLE = {Degrees of rigidity for {S}ouslin trees},
JOURNAL = {Journal of Symbolic Logic},
FJOURNAL = {Journal of Symbolic Logic},
VOLUME = {74},
YEAR = {2009},
NUMBER = {2},
PAGES = {423--454},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E05},
MRNUMBER = {2518565 (2010i:03049)},
MRREVIEWER = {Stefan Geschke},
URL = {http://wp.me/p5M0LV-3A},
doi = {10.2178/jsl/1243948321},
eprint = {math/0602482},
archivePrefix = {arXiv},
primaryClass = {math.LO},
file = F,
}

We investigate various strong notions of rigidity for Souslin trees, separating them under Diamond into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under Diamond that there is a group whose automorphism tower is highly malleable by forcing.

# Canonical seeds and Prikry trees

• J. D. Hamkins, “Canonical seeds and Prikry trees,” Journal of Symbolic Logic, vol. 62, iss. 2, pp. 373-396, 1997.
@article {Hamkins97:Seeds,
AUTHOR = {Hamkins, Joel David},
TITLE = {Canonical seeds and {P}rikry trees},
JOURNAL = {Journal of Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {62},
YEAR = {1997},
NUMBER = {2},
PAGES = {373--396},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E40 (03E05 03E55)},
MRNUMBER = {1464105 (98i:03070)},
MRREVIEWER = {Douglas R.~Burke},
DOI = {10.2307/2275538},
URL = {http://jdh.hamkins.org/seeds},
}

Applying the seed concept to Prikry tree forcing $\mathbb{P}_\mu$, I investigate how well $\mathbb{P}_\mu$ preserves the maximality property of ordinary Prikry forcing and prove that $\mathbb{P}_\mu$ Prikry sequences are maximal exactly when $\mu$ admits no non-canonical seeds via a finite iteration.  In particular, I conclude that if $\mu$ is a strongly normal supercompactness measure, then $\mathbb{P}_\mu$ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. H. Woodin’s.