Most of the active Jewish geocentrists are Lubavitchers. In contrast to the Christian geocentrists who have as their main impetus for geocentrism various Biblical verses, 2 the Jewish geocentrists seem to in a large part be motivated to preserve the correctness of certain statements by Maimonides. In particular, they defend the cosmology as set out by the Maimonides in the Mishne Torah. I am puzzled by these apologetics for two reasons: First, there does not appear to be any similar attempt to defend incorrect medical statements by Maimonides. Second, there’s no theological need even among charedim to believe that Maimonides was infallible. Prior to this, I have seen attempts to argue for what amount to infallibility of the Talmudic authors but had not previously encountered such attempts where later authors such as Maimonides were concerned.3
Like the Christian geocentrists, the Jewish geocentrists prefer two lines of argument which are contradictory and yet are used in tandem. First, they argue that under General Relativity one can always construct a coordinate system with a given point as stationary so one might as well assume that the Earth is at the center.4 This argument has been sometimes called “weak geocentrism.
The second argument, “strong geocentrism,” is the claim that the Earth is at the center of the universe in a meaningful fashion; that is, there are lines of evidence which only make sense if the Earth is at the center of the universe. How both these claims can be made simultaneously is not clear to me. Apparently, studying Talmud on a daily basis does not help ones logical skills as much as one might hope. As with the Christian geocentrists, the Jewish geocentrists have a strong anti-science bent; they either insinuate or state outright that there exists a massive conspiracy within the scientific establishment to discredit geocentrism. Like their Christian counterparts, the Jewish geocentrists credit the Copernican system with inspiring later heresy such as evolution.
On the whole, the Jewish geocentrists in thoughts and arguments are very similar to the Christian geocentrists. As with the Jewish and Christian anti-evolutionists it is not obvious to me whether there is cross-fertilization or independent arrival at the same ideas. I suspect there is cross-fertilization but the mechanism is unclear. More research is needed.
While Catholic geocentrists also exist they are rarer than their Protestant counterparts. Geocentrism is in fact not the most extreme extant cosmological belief; there are a handful of extant flat-earthers especially among Islamic groups. See for example this video of a debate on Iraqi television about whether the Earth is flat.
2. Such as Joshua 10 in which the sun stands still.
3. Orthodox apologists use a variety of different arguments to defend against apparent errors of fact in the Talmud. One of the more transparent attempts is to claim that the nature of the universe has changed since the Talmudic time. This claim shows up in a variety of different forms including explaining the Talmud’s apparent confusion about the human female menstrual cycle. My impression is that many such claims are pro forma; I have yet to find anyone who was been willing to state explicitly that they think the basics of human anatomy have changed in the last 2000 years.
4. I’m not sure this is in fact strictly true. I’d appreciate input on this subject from people who understand general relativity. In fact, this weak geocentrism is not sufficient to save Maimonides's cosmology since he has the various planets in circular orbits about the Earth and has the sun in a circular orbit about the earth (along with a few epicycles). This contradicts observed data even if the Earth were stationary.