On computing homology gradients over finite fields

Abstract

Abstract. Recently the so-called Atiyah conjecture about l 2 -Betti numbers has been disproved. The counterexamples were found using a specific method of computing the spectral measure of a matrix over a complex group ring. We show that in many situations the same method allows to compute homology gradients, i.e. generalizations of l 2 -Betti numbers to fields of arbitrary characteristic. As an application we point out that (i) the homology gradient over any field of characteristic different than 2 can be an irrational number, and (ii) there exists a finite CW-complex with the property that the homology gradients of its universal cover taken over different fields have infinitely many different values.

L2-Betti Numbers and their Analogues in Positive Characteristic

“…If G is an amenable group we have constructed rk G as a Sylvester matrix rank function not only on C [G] but also on K [G] for every field K. In particular, we can formulate an analogue of Atiyah's question in characteristic p: is it true that rk G takes only rational values as a Sylvester matrix rank function on F p [G]? This question was considered in [45] where it was shown that for every real number r there exists an amenable group G such that r ∈ A Fp (G). Again, as in the case of similar examples in characteristic 0, the examples of groups from [45] have finite subgroups of unbounded order.…”

“…This question was considered in [45] where it was shown that for every real number r there exists an amenable group G such that r ∈ A Fp (G). Again, as in the case of similar examples in characteristic 0, the examples of groups from [45] have finite subgroups of unbounded order.…”

On an analogue of L2-Betti numbers for finite field coefficients and a question of Atiyah

“…In the second part we rebuild the computational tool from [GS14] for characteristic 2. Then corresponding to L 2 -Betti numbers of normal coverings we show, for any real number r, the construction of a finitely generated amenable group and an associated F 2 [G] matrix, whose kernel has Følner dimension r.…”

“…To showcase some calculations of the Følner dimension previously defined, we amend the results of [GS14] with the case of F 2 , the field of 2 elements.…”

“…The theorem we have just proven allows us to compute the dimension of the kernel by way of adjacency operators on certain labeled graphs. In the calculation for the main theorems of this section, we need one class of labeled graphs, simply called nice graphs following the idea in [GS14].…”