桃色视频

1. Jessica Anzanello: Nearly fixed-point-free elements in permutation groups with two orbits

   A classical theorem of Jordan states that every non-trivial finite transitive group contains a derangement, that is, an element with no fixed points. A deep result of Fein, Kantor, and Schacher shows that one can even find such an element of prime-power order. In this talk, I will discuss a natural extension of this result to permutation groups G with two orbits. In this setting, we show that either G contains a derangement, or it contains an element of prime-power order with exactly one fixed point. As a corollary, we obtain that if the two orbits have size a and b, with a + b > 2 and (a, b 鈭� 1) = (a 鈭� 1, b) = 1, then G has a derangement. This confirms and generalizes a conjecture of Ellis and Harper. This is joint work with Sean Eberhard. slides

2. Adrian Beker: The R茅nyi entropy of the order of a random permutation

   The study of the order of a random permutation is a fundamental problem in probabilistic group theory. Its global behaviour is well understood: a classical result of Erd艖s and Tur谩n shows that the logarithm of the order is asymptotically normally distributed. We consider two local questions: what is the most probable order of a random permutation of {1, . . . , n}, and with what probability does it occur? What is the probability that two independent random permutations have equal orders? The former goes back to work of Erd艖s and Tur谩n, while the latter was recently studied by Acan, Burnette, Eberhard, Schmutz and Thomas. We give an essentially complete answer to these questions by analysing the associated concept of R茅nyi entropy. Our results are quantitatively optimal and reveal a tight connection between the asymptotic behaviour of the quantities in question and number-theoretic properties of n. slides

3. Ohad Sheinfeld: Improved covering results and intersection theorems in symmetric groups, via hypercontractivity

   In this talk, we present two seemingly unrelated results on the symmetric group S_n. A well-known problem asks for a characterization of subsets A of A_n whose square A² covers the entire alternating group A_n. We show that if A is a conjugacy class of density at least exp(鈭�n^2/5 鈭� 蔚), then A_n = A². This improves upon a seminal result by Larsen and Shalev, who obtained the same result with 1/4 in the double exponent.

   The second problem is an analogue of the Erd艖s鈥揝贸s forbidden intersection problem for families of permutations. A family F of permutations is called non-(t 鈭� 1)-intersecting if any two permutations in F do not agree on exactly (t 鈭� 1) values. We prove that for t < 鈭�n / log(n), the maximal size of a non-(t 鈭� 1)-intersecting family is (n 鈭� t)!, which improves upon the result of Kupavskii and Zakharov, who proved the same for t = O虄(n^1/3).

   Both results can be interpreted as bounds on the size of independent sets in certain normal Cayley graphs over symmetric groups. The primary tool we use in both proofs to establish these bounds are the recently obtained hypercontractive inequalities for global functions, developed by Keller, Lifshitz, and Marcus, as well as by Keevash and Lifshitz. The talk is based on joint work with Nathan Keller and Noam Lifshitz. slides

Gong Show Session 3: Cayley graphs, expansion, and isoperimetry

1. Marco Barbieri: Strong convergence of random permutational representations

   A family of d-regular graphs is called an expander family if their adjacency matrices exhibit a spectral gap between the trivial eigenvalue d and the rest of the spectrum. This seemingly simple condition has far-reaching consequences, from rapid mixing of random walks to strong metric properties. On a seemingly different note, a sequence of random matrices is said to strongly converge to an operator if, with high probability, the norms of these matrices converge to the operator norm of the limiting object. This notion has recently found powerful applications in expansion problems, both in graph theory and in hyperbolic geometry. In this short talk, we will explore the connection between these two notions, and we will show how strong convergence of random permutational representations of symmetric groups leads to probabilistic constructions of expanders. This is joint work with Urban Jezernik. slides

2. Nathan Deloire: Isoperimetry of diagonal products of permutational wreath products

   While a construction of groups with prescribed isoperimetric behaviour is known in the exponential growth setting, much less is understood for groups of intermediate growth.

   In 2021, J茅r茅mie Brieussel and Tianyi Zheng constructed finitely generated groups with prescribed isoperimetry for a large class of isoperimetric profiles, or equivalently return probabilities. These groups are constructed using the notion of 鈥渄iagonal product鈥� of marked wreath products 螕_m 鈮赌 Z, with (螕_m)_{m 鈭� N} a sequence of finite groups, and yield groups of exponential growth. The key step of the proof is the evaluation of the asymptotic isoperimetry of such groups.

   In my research, I am adapting this idea in order to construct intermediate growth groups with prescribed isoperimetry by studying the asymptotic isoperimetry of the diagonal products of groups of intermediate growth. The groups of which I am studying the diagonal product are permutational wreath products 螕_m 鈮赌_X G_蝇, with (螕_m)_{m 鈭� N} a sequence of finite groups and G_蝇 a Grigorchuk group, and whose intermediate growth was proved in 2014 by Bartholdi and Erschler. slides

3. Cosmas Kravaris: Do the symmetric groups embed into the Banach space L₁?

   Kassabov鈥檚 theorem states that the symmetric groups admit bounded size generating sets whose Cayley graphs form expanders. When the generating set is a cycle and a transposition, it is easy to check that the Cayley graph is not an expander. By a classical argument of Gromov, expander graphs do not embed coarsely into L₁. On the other hand, we show that the Cayley graph generated by a cycle and a transposition embeds into L₁ with bi-Lipschitz distortion < 1000. In sharp contrast to the study of infinite groups, we see that the large-scale metric geometry of a finite group depends heavily on the choice of generators. slides

4. Elena Maini: Diameter bounds for Cayley graphs of arbitrary finite groups

   Investigating the growth of a finite group involves looking at its Cayley graphs. For example, it is interesting to estimate the maximum diameter of a connected Cayley graph of a finite group G, which is usually referred to as the diameter of G. I will present a new result stating that, given an arbitrary finite group G, the diameter of G can be controlled provided that the normal abelian sections of G have small exponent and the composition factors of G have small diameter. I will then discuss some surprising consequences of this fact. This is joint work with Sean Eberhard, Luca Sabatini, and Gareth Tracey. slides

5. Luca Sabatini: Cayley graphs of quasirandom groups

   Building on the recent work of Golsefidy鈥揝rinivas, we present new expansion properties of abstract quasirandom groups. slides

6. Yotam Shomroni: Independent words are freely separated

   Every word w in a free group F defines, for each finite group G, a G-valued random variable, by substituting independent, uniformly random G-elements for the letters. If w_1 and w_2 share no letters, they define independent random variables. We prove the converse: if w_1 and w_2 are independent for every finite G, then they share no letters when written in some basis of F. In fact, it suffices that w_1 and w_2 are independent in the symmetric groups S_n: we show that the correlation between the events of w_i fixing some point is non-negative for all large enough n, and positive unless the words share no letters (in some basis). Joint work with Noam Ta-Shma. slides