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In this article, we study the asymptotic behaviour of conjugacy separabilityfor wreath products of abelian groups. We fully characterise the asymptoticclass in the case of lamplighter groups and give exponential upper and lowerbounds for generalised lamplighter groups. In the case where the base group isinfinite, we give superexponential lower and upper bounds. We apply our resultsto obtain lower bounds for conjugacy depth functions of various wreath productsof groups where the acting group is not abelian.
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Wreath-like products of groups and their von Neumann algebras I: W^∗-superrigidity
We introduce a new class of groups called {\it wreath-like products}. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have Kazhdan's property (T). In this paper, we prove that any group $$G$$ in a natural family of wreath-like products with property (T) is W$^*$-superrigid: the group von Neumann algebra $$\text{L}(G)$$ remembers the isomorphism class of $$G$$. This allows us to provide the first examples (in fact, $$2^{\aleph_0}$$ pairwise non-isomorphic examples) of W$^*$-superrigid groups with property (T).
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- Award ID(s):
- 1854194
- PAR ID:
- 10510958
- Publisher / Repository:
- Princeton University; Institute for Advanced Study
- Date Published:
- Journal Name:
- Annals of Mathematics
- Volume:
- 198
- Issue:
- 3
- ISSN:
- 0003-486X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Srinivasan, Srikanth (Ed.){"Abstract":["We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: \r\n1) We consider read-once group-products over a finite group G, i.e., tests of the form ∏_{i=1}^n (g_i)^{x_i} where g_i ∈ G, a special case of read-once permutation branching programs. We give generators with optimal seed length c_G log(n/ε) over any p-group. The proof uses the small-bias plus noise paradigm, but derandomizes the noise to avoid the recursion in previous work. Our generator works when the bits are read in any order. Previously for any non-commutative group the best seed length was ≥ log n log(1/ε), even for a fixed order.\r\n2) We give a reduction that "lifts" suitable generators for group products over G to a generator that fools width-w block products, i.e., tests of the form ∏ (g_i)^{f_i} where the f_i are arbitrary functions on disjoint blocks of w bits. Block products generalize several previously studied classes. The reduction applies to groups that are mixing in a representation-theoretic sense that we identify.\r\n3) Combining (2) with (1) and other works we obtain new generators for block products over the quaternions or over any commutative group, with nearly optimal seed length. In particular, we obtain generators for read-once polynomials modulo any fixed m with nearly optimal seed length. Previously this was known only for m = 2.\r\n4) We give a new generator for products over "mixing groups." The construction departs from previous work and uses representation theory. For constant error, we obtain optimal seed length, improving on previous work (which applied to any group). \r\nThis paper identifies a challenge in the area that is reminiscent of a roadblock in circuit complexity - handling composite moduli - and points to several classes of groups to be attacked next."]}more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4007/annals.2023.198.3.6
