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publications
New hyperfinite subfactors with infinite depth
Published in arXiv:2509.00227. Submitted, 2025
We construct new hyperfinite subfactors with Temperley-Lieb-Jones (TLJ) standard invariant and Jones indices between \(4\) and \(3 + \sqrt{5}\). Our subfactors occur at all indices between \(4\) and \(5\) at which finite depth, hyperfinite subfactors exist. The presence of hyperfinite subfactors with TLJ standard invariant, and hence of infinite depth, at these special indices comes as a surprise. In particular, we obtain the first example of a hyperfinite subfactor with “trivial” (i.e. TLJ) standard invariant and integer index \(>4\), namely with index \(5\). This is achieved by constructing explicit families of nondegenerate commuting squares of multi-matrix algebras based on exotic inclusion graphs with appropriate norms. We employ our graph planar algebra embedding theorem, a result of Kawahigashi and other techniques to determine that the resulting commuting square subfactors have indeed TLJ standard invariant. We also show the existence of a hyperfinite subfactor with trivial standard invariant and index \(3 + \sqrt{5}\). It is interesting that this index features at least two distinct infinite depth, irreducible hyperfinite subfactors and several finite depth ones as well. Our work leads to the conjecture that every Jones index \(>4\) of an irreducible, hyperfinite (finite depth) subfactor can be realized by one with TLJ standard invariant.
Recommended citation: Dietmar Bisch, Julio Cáceres. (2025). "New hyperfinite subfactors with infinite depth" arXiv:2509.00227
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Graph planar algebra embeddings and infinite depth subfactors
Published in International Journal of Mathematics, 2025
Subfactors of the hyperfinite II\(_1\) factor with “exotic” properties can be constructed from nondegenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones’ graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley-Lieb-Jones standard invariant and index \(\frac{5+\sqrt{13}}{2}\), i.e. the index of the Haagerup subfactor. We are led to a conjecture pertaining to Jones indices of irreducible, hyperfinite subfactors.
Recommended citation: Dietmar Bisch, Julio Cáceres. (2025). "Graph planar algebra embeddings and infinite depth subfactors" International Journal of Mathematics , Vol. 36, No. 13
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The Carpenter property for subfactors of uniform infinite index
Published in arXiv:2611.02006, 2026
We show that if \(N\subset M\) is an inclusion of II\(_1\) factors of uniform infinite index, for instance an irreducible inclusion of infinite index, then every positive contraction \(b\in N\) is of the form \(E_N(p)\) for a projection \(p\in M\). This gives the exact version of Popa’s asymptotic result in the ultrapower, and it is a subfactor analogue of the Carpenter problem for masas. In particular, Popa’s relative dimension set of such an inclusion is \([0,1]\), so the set of expectations of projections does not distinguish inclusions without corners of finite index. As applications, the Carpenter property holds for \(N\subset N\rtimes G\) whenever an infinite group \(G\) acts outerly on a II\(_1\) factor \(N\), and for ICC groups \(H\le G\) the inclusion \(L(H)\subset L(G)\) has it exactly when \([G:H]=\infty\), confirming a recent conjecture. We also give examples showing that the Carpenter property does not imply uniform infinite index, and that infinite index does not imply the Carpenter property.
Recommended citation: Dietmar Bisch, Julio Cáceres. (2026). "The Carpenter property for subfactors of uniform infinite index" arXiv:2611.02006
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The relative dimension set of an extremal subfactor of finite index
Published in arXiv:2611.02336, 2026
Popa’s relative dimension set \(\Lambda(M,N)\) of an inclusion of II\(_1\) factors \(N\subset M\) consists of the scalars \(\alpha\) for which some projection \(p\in M\) satisfies \(E_N(p)=\alpha 1\). Popa computed it for index at most \(4\), and for index \(d>4\) outside the middle interval \((t_d,1-t_d)\), where \(t_d(1-t_d)=d^{-1}\); whether the middle interval meets \(\Lambda(M,N)\) for every irreducible inclusion of index \(d>4\) has remained open. We show that for every extremal inclusion of finite index \(d>4\), the whole interval \([t_d,1-t_d]\) is contained in \(\Lambda(M,N)\). Consequently, \(\Lambda(M,N)\) depends only on the index for all extremal inclusions of finite index, in particular for all irreducible ones. Extremality cannot be dropped: for \(d>4\), Popa’s locally trivial inclusions are exactly those whose relative dimension set misses the middle interval. Other non-extremal inclusions can contain the whole middle interval in their relative dimension set, or miss part of it.
Recommended citation: Julio Cáceres. (2026). "The relative dimension set of an extremal subfactor of finite index" arXiv:2611.02336
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talks
Subfactors
Published:
New hyperfinite subfactors with infinite depth
Published:
Graph planar algebra embeddings and infinite depth subfactors
Published:
Planar Algebras
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New hyperfinite subfactors with infinite depth
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New hyperfinite subfactors with infinite depth
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New hyperfinite subfactors with infinite depth
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New hyperfinite subfactors with infinite depth
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teaching
Math 1301
Teaching Assistant, Vanderbilt University, Department of Mathematics, Fall 2020
Math 1301
Instructor, Vanderbilt University, Department of Mathematics, Spring 2021
Math 1300
Instructor, Vanderbilt University, Department of Mathematics, Fall 2021
Math 1201
Teaching Assistant, Vanderbilt University, Department of Mathematics, Spring 2022
Math 1300
Instructor, Vanderbilt University, Department of Mathematics, Fall 2022
Math 1301
Teaching Assistant, Vanderbilt University, Department of Mathematics, Spring 2023
Math 1300
Teaching Assistant, Vanderbilt University, Department of Mathematics, Fall 2023
Math 1301
Teaching Assistant, Vanderbilt University, Department of Mathematics, Spring 2024
Math 2300
Instructor, Vanderbilt University, Department of Mathematics, Fall 2024
Math 2610
Instructor, Vanderbilt University, Department of Mathematics, Spring 2025
Math 2400
Instructor, Vanderbilt University, Department of Mathematics, Fall 2025
Math 3100
Instructor, Vanderbilt University, Department of Mathematics, Spring 2026
Math 3100
Instructor, Vanderbilt University, Department of Mathematics, Fall 2026